CONTRIBUCION AL ESTUDIO DE EMBALDOSADOS APERIODICOS: … · crystal" or quasicrystal. After this...

UNIVERSIDAD DE CHILE

FACULTAD DE CIENCIAS FISICAS Y MATEMATICAS

DEPARTAMENTO DE INGENIERIA MATEMATICA

CONTRIBUCION AL ESTUDIO DE EMBALDOSADOS

APERIODICOS: TEORIA DE TRANSLACION Y

PROPIEDADES ESTADISTICAS

TESIS PARA OPTAR AL GRADO DE DOCTOR EN CIENCIAS DE LA INGENIERIAMENCION MODELACION MATEMATICA

EN COTUTELA CON LA UNIVERSIDAD DE NIZA-SOPHIA ANTIPOLIS

JOSE EDUARDO ALISTE PRIETO

PROFESOR GUIA :

ALEJANDRO MAASS SEPULVEDA

MIEMBROS DE LA COMISION:

JEAN-RENE CHAZOTTES

JEAN-MARC GAMBAUDO

JAROSLAW KWAPISZ

SERVET MARTINEZ AGUILERA

SAMUEL PETITE

JAIME SAN MARTIN ARISTEGUI

SANTIAGO - CHILE

JUNIO 2009

RESUMEN DE LA TESIS PARA OPTAR AL GRADO DE DOCTOR

EN CIENCIAS DE LA INGENIERIA MENCION MODELACION MATEMATICA

POR : JOSE EDUARDO ALISTE PRIETO

FECHA : 23 JUNIO 2009

PROF. GUIA: ALEJANDRO MAASS SEPULVEDA

CONTRIBUCION AL ESTUDIO DE EMBALDOSADOS APERIODICOS:

TEORIA DE TRANSLACION Y PROPIEDADES ESTADISTICAS

Esta tesis se enmarca en el estudio de los sistemas dinamicos asociados a la envoltura de un

embaldosado o conjunto de Delone repetitivo aperiodico del espacio euclidiano con frecuencia

uniforme de motivos.

En la primera parte se estudia la accion por traslacion sobre la envoltura en el caso lin-

ealmente repetitivo. Mas precisamente, se construyen sistemas de torres para la envoltura,

refinando la construccion original de Bellissard, Benedetti y Gambaudo. Este refinamiento

permite demostrar que las matrices de transicion del sistema de torres estan uniformemente

acotadas en dimension y en norma. Posteriormente, se estudia la tasa de convergencia de

la frecuencia de un motivo a su promedio ergodico y se obtiene una estimacion de ella en

terminos del sistema de torres. A partir de esta estimacion se obtiene una condicion sufi-

ciente para que la tasa de convergencia sea optima. Ademas, utilizando un lema de mezcla

tipo Perron-Frobenius se da una nueva demostracion de un resultado de Lagarias y Pleasants

sobre la tasa de convergencia de la frecuencia de motivos. Este es un trabajo conjunto con

Alvaro Coronel.

En la segunda parte se dan los fundamentos para una teorıa de translacion de los homeo-

morfismos de la envoltura de un conjunto de Delone de la recta real. Se define y estudia el

conjunto de translacion de los homeomorfismos homotopicos a la identidad que preservan la

orientacion. Uno de los resultados asegura la existencia de un unico numero de traslacion para

homeomorfismos que no poseen puntos fijos. En paralelo con la teorıa clasica de Poincare, se

estudia la existencia de una semi-conjugacion entre un homeomorfismo sin puntos fijos y la

translacion por su numero de translacion. Se define una nocion alternativa de numero racional

(que llamaremos T -racional) y se establece que bajo una hipotesis de acotamiento sobre el

co-ciclo de desplazamiento, los homeomorfismos con un numero de translacion T -irracional

son semiconjugados a una translacion, en paralelo con el resultado clasico de Poincare.

Agradecimientos

En primer lugar quiero agradecer a mis directores de tesis, Alejandro Maass y Jean-Marc

Gambaudo, por su constante apoyo y su guıa a largo de esta tesis.

Me siento muy honrado de haber contado con Jaroslaw Kwapisz y Patrice Le Calvez

como referees de mi tesis. En particular, agradezco a Jaroslaw Kwapisz por productivas

discusiones. Agradezco tambien a Jean-Rene Chazottes, Servet Martinez, Jaime San Martın

y Samuel Petite por aceptar ser parte del jurado de mi defensa.

Agradezco a Samuel, Daniel, Cristobal y Mario por diversas discusiones sobre mi trabajo

de tesis y mas generalmente sobre los sistemas dinamicos. Hago mencion especial a Samuel

por su generosidad y hospitalidad en mis visitas a Amiens.

Agradezco a CONICYT por financiar mis estudios de Doctorado en Chile, al programa

ARCUS por financiar mi estadıa en Parıs y al proyecto ECOS C03EC03 por financiar mi

estadıa de cotutela en Niza. Durante mi estadıa en Niza agradezco el apoyo del gobierno

frances a traves del CROUS de Niza y la CAF de los alpes maritimos, asi como el finan-

ciamiento para participar en diversas escuelas y workshops otorgados por el ANR Crystal

Dyn, el CIRM en Marsella entre otros. Finalmente, agradezco el apoyo dados por el Nucleo

Milenio P04-069-F y el proyecto Basal del Centro Modelamiento Matematico en la etapa final

de redaccion de este trabajo.

Tuve la suerte de compartir durante mi estadıa en Niza y Parıs con diversas personas a las

cuales quisiera agradecer: Alejandra, Angela, Daniel, Guillermo, Juan Carlos, Mara, Nico,

Marcello, Laura, Marc, Xavier, Delphine, Pierre, Vanessa, Eduardo, Stefania, Elo, Maıwenn,

Sigo, Julie, Laetitia, Ali, Cristobal, Cathy, Claudia, Eduardo, Fernando, Giorgo, Mario,

Tamara, Pamela y Philipp.

Finalmente quisiera agradecer a mi familia por todo su apoyo durante estos anos, en

particular a mi madre, por ser vivo ejemplo de entereza y perseverancia, a mi hermano

Rodrigo por haberse quedado entre nosotros y por sobre todo a mi hijo Nicolas por ese amor

incondicional que me entrega y toda la alegrıa que me provoca verlo crecer. Este ultimo ano

lo he compartido con Tomke, cuyo carino, paciencia y aliento fueron cruciales para llevar a

cabo la redaccion de este trabajo.

3

para mi hijo Nicolas, que la vida se te de llena de felicidad y aventuras

Contents

List of Figures 7

1 Introduction 8

1.1 The discovery of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Linearly repetitive Delone sets and their associated dynamical systems . . . 11

1.3 Results of this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Fast convergence to patch frequencies for linearly repetitive Delone sets 13

1.3.2 Translation numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Towards a Poincare-like classification . . . . . . . . . . . . . . . . . . 19

2 Tilings, Delone sets and Delone dynamical systems 21

2.1 Delone sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Delone systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 From tilings to punctured tilings and Delone sets . . . . . . . . . . . 27

2.3.2 Voronoi tilings: from Delone sets to tilings . . . . . . . . . . . . . . . 27

2.4 The canonical transversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Entrance and return times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Unique ergodicity versus uniform pattern frequencies . . . . . . . . . . . . . 31

2.7 Laminations and Rd-solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Delone systems are Rd-solenoids . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.8.1 Local verticals and local transversals . . . . . . . . . . . . . . . . . . 37

2.8.2 Invariant measures and transverse invariant measures . . . . . . . . . 38

3 Hierarchy of aperiodic Delone sets 39

3.1 Boxes and canonical parametrizations . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Box decompositions and derived tilings . . . . . . . . . . . . . . . . . . . . . 40

3.3 Voronoi box decompositions and Derived Voronoi tilings . . . . . . . . . . . 42

3.4 Tower systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5

Contents

3.5 Tower systems for linearly repetitive Delone sets . . . . . . . . . . . . . . . . 48

4 On the rate of convergence of patch frequencies for linearly repetitive Delone

sets 53

4.1 Motivations and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Estimating the error term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 A proof of Lagarias and Pleasants Theorem . . . . . . . . . . . . . . . . . . 59

5 Pattern-equivariance and translation numbers for maps in the real-line 62

5.1 Introduction: short range potentials and pattern-equivariant functions . . . . 62

5.2 Return times and Poincare maps for local verticals in Delone spaces . . . . . 65

5.3 Maps on Delone systems that are homotopic to the identity . . . . . . . . . 67

5.4 Translation sets for self-maps homotopic to the identity . . . . . . . . . . . . 69

6 Towards a Poincare’s Theorem 78

6.1 Replacing the rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2 Γ-semiconjugacies and ρ-bounded maps . . . . . . . . . . . . . . . . . . . . . 81

6.3 Towards a Poincare Theorem for ρ-bounded homeomorphisms in H++(ΩX) . 85

6.3.1 The Γ-rational case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3.2 The Γ-irrational case . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Appendix: a tractable condition for ρ-boundedness . . . . . . . . . . . . . . 95

Bibliography 97

Index 101

6

List of Figures

1.1 The diffraction image of the Shechtman quasicrystal. . . . . . . . . . . . . . 9

1.2 The set of 13 aperiodic Wang tiles found by Culik. . . . . . . . . . . . . . . . 10

1.3 Penrose tiles with local rules enforced by cuts in the edges. . . . . . . . . . . 11

1.4 A region of a Penrose tiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 A punctured Penrose tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 A region of Voronoi tiling associated to the set of punctures of a punctured

Penrose tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 yi − yj does not depend on Y . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Derived tiling of the Voronoi Box decomposition with base equal to the cylinder

of a white-star in the Penrose tiling . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 The derived tiling of a box decomposition zoomed out of the canonical Penrose

tiling obtained by the ”surgery” process to the Voronoi box decomposition

illustrated in Figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Description of the patches of different levels using the box decomposition of

Figure 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 Construction of the Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 A Fib-equivariant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 First return time and first return maps for Fibonacci quasicrystals. Here

V = Y ∈ ΩFib : Y ∩ [−L,L] = −L,L. . . . . . . . . . . . . . . . . . . . 66

5.4 Proof of Theorem 5.4.3: On the left: j`+1 = j` + kn(Y`). On the right: j`+1 =

j` + kn(Y`) + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7

1 Introduction

1.1 The discovery of quasicrystals

Let Rd be the Euclidean space equipped with the usual Euclidean norm. A discrete subset

X of Rd is called a Delone set if it is uniformly discrete (there exists r > 0 such that every

closed ball of radius r intersects X in at most one point) and relatively dense (there exists

R > 0 such that every closed ball of radius R intersects X in at least one point). Delone sets

arise naturally as mathematical models for the description of solids. In this modelization, the

solid is supposed to be infinitely extended and its atoms are represented by points. These

atoms interact through a potential (for example a Lennard-Jones potential). For a given

specific energy, Delone sets are good candidates to describe the ground state configuration:

uniform discreteness corresponds to the existence of a minimum distance between atoms due

to the repulsion forces between nuclei, and relative density corresponds to the fact that empty

regions cannot be arbitrarily big because of the contraction forces. In perfect crystals, atoms

are ordered in a repeating pattern extending in all three-dimensions and can be modeled

by lattices in R3 (which are clearly Delone sets). These were classified in the nineteenth

century by the french physicist A. Bravais, who determined that there are exactly 14 different

repeating patterns up to scale and rigid motions.

The X-ray diffraction image of an alloy in a solid state can be a very wild object. However,

when the solid is a perfect crystal, this image consists of sharp dots (pure point spectrum),

and its symmetries are the symmetries of the periodic pattern. Thus they belong to the

symmetry group of one of the 14 Bravais patterns. In the eighties, D. Shechtman and his

team [Shecthman 84] observed a metallic solid (an alloy of aluminum and manganese) having

long-range order. This means that the spots on its diffraction image are as sharp as in a

crystal and every finite configuration repeats itself in every big enough region (see Figure

1.1). The key feature observed by Shechtman was that the diffraction image of this solid also

exhibited a 5-fold symmetry, which is forbidden by Bravais classification. Therefore this solid

8

1 Introduction

was not a perfect crystal. Nowadays, the term “crystal” means any material having sharp

spots in his diffraction image, and the solid found by Shechtman is known as a “quasiperiodic

crystal” or quasicrystal . After this discovery, more than a hundred quasicrystals have been

discovered, most of them, alloys containing aluminium.

Figure 1.1: The diffraction image of the Shechtman quasicrystal.

In mathematical terms, long-range order can be defined as follows. Let X be a Delone set.

A patch of X is a set of the form X ∩B, where B is a closed ball in Rd. If the radius of B

is equal to S, then the patch is called a S-patch . The Delone set X has finite type if for

every S > 0 there is a finite number of S-patches modulo translation. The Delone set X is

said to be repetitive if every patch of X repeats itself all along X, i.e. there exists M > 0

such that every ball of radius M contains a patch that after a translation coincides with the

original patch. It follows that repetitive Delone sets have finite type. Lattices are repetitive

Delone sets, but there exist repetitive Delone sets that are aperiodic, i.e. the set coincides

with none of its translations.

A tiling is a countable cover of Rd by closed sets, called tiles, having pairwise disjoint in-

teriors. A patch of a tiling is a finite subset of it. The notions of repetitivity and aperiodicity

extend to tilings.

Aperiodic tilings have appeared in the context of logic before Shecthmann’s discovery.

A Wang tile is a 1 × 1 square with each edge colored, and a Wang system is a finite

collection of Wang tiles. We say that a Wang system tiles the plane if there exists a tiling

of the plane satisfying the following:

9

1 Introduction

• Each tile is a copy of a tile in the Wang system.

• Tiles intersect full-edge to full-edge.

• Any two overlapping edges have the same color.

The domino problem is to know, for a given Wang system, whether the system tiles the

plane or not. In [Wang 61], H. Wang analyzed the decidability of this problem and showed

that it is equivalent to the existence of a periodic tiling for every Wang system that tiles

the plane. This latter assumption is now known as Wang’s conjecture and was proved

wrong in 1966 [Berger 66], when R. Berger constructed a Wang system, consisting of 20, 426

Wang tiles, that tiles the plane but for which no periodic tiling exists. A Wang system that

tiles the plane only in an aperiodic way is called aperiodic. In the following years, several

people simplified Berger’s construction. Particularly important is the work of [Culik 96], who

constructed an aperiodic Wang system with 13 tiles. This is the smallest known example of

an aperiodic Wang system. Figure 1.2 shows a picture of the Wang system constructed by

Culik.

Figure 1.2: The set of 13 aperiodic Wang tiles found by Culik.

The domino problem can be generalized to polygons and local rules that state whether

two parallel edges in a finite system of polygons can be glued in an allowed tiling or not. In

this context, Penrose found several systems of polygons and local rules enforcing aperiodicity.

For instance, the set of Penrose rhombs is a set of only 10 polygons (2 if we count up

to rotation, see Figure 1.3). The tilings built with these systems are commonly known as

Penrose tilings . The remarkable fact about these tilings is that they exhibit the same

5-fold symmetry that appears in the diffraction image of the Shechtman quasicrystal. This

can be seen as a motivation for using aperiodic tilings to model quasicrystals. In Figure 1.4

a region of a Penrose tiling is shown.

10

1 Introduction

Figure 1.3: Penrose tiles with local rules enforced by cuts in the edges.

Figure 1.4: A region of a Penrose tiling.

1.2 Linearly repetitive Delone sets and their associated

dynamical systems

A Delone set X is called linearly repetitive if there exists a constant L > 0 such that

for every R > 0 the ball of radius LR contains a translated of every possible R- patch.

Linearly repetitive Delone sets were introduced by Lagarias and Pleasants in [Lagarias 03]

as a possible notion of “perfectly ordered quasicrystals”. They contain the class of aperiodic

sets coming from self-similar tilings with the unique composition property (see [Solomyak 98,

Lemma 2.3]). In particular, the Penrose tilings described above are linearly repetitive.

The theory of dynamical systems has proved to be a powerful tool in the study of aperiodic

Delone sets and tilings. This approach is based on the following construction: we denote by

Γ the group of rigid translations of the Euclidean space. This group acts in a natural way

on Delone sets; if X is a Delone set and v ∈ Rd then ΓvX is defined by

ΓvX = X − v = x− v : x ∈ X.

11

1 Introduction

Let X be an aperiodic Delone set of finite type in Rd, and consider its orbit O(X) = X−x :

x ∈ Rd endowed with a metrizable topology that says that two Delone sets in O(X) are close

if they coincide, after a small translation, in a big ball around the origin. We remark that

the aperiodicity of X implies that O(X) is not complete. Hence, we define the continuous

hull of X, which will be denoted by ΩX , to be the completion of O(X). It is a standard

fact that ΩX is a compact metric space (see [Rudolph 89]). Moreover, the elements of ΩX

are Delone sets having the same patches as X. Thus, there is a natural action of Γ on ΩX .

This action is continuous, and hence the pair (ΩX ,Γ) is a topological dynamical system .

It is known that the space ΩX is locally homeomorphic to the product of a Cantor set and

Rd (see Section 2.8 for more details). This means that for each Y ∈ ΩX and each closed ball

BS ⊆ Rd of radius S around 0, the cylinder set

CY,S = Z ∈ ΩX | Z ∩BS = Y ∩BS

is a Cantor set. We say that a clopen subset of CY,S is a local vertical (see Section 2.4 for

more details).

Moreover, there is a relationship between geometrical and combinatorial properties of an

aperiodic Delone set X of finite type and dynamical properties of the system (ΩX ,Γ). For

instance, a result of [Gottschalk 44] implies that (ΩX ,Γ) is minimal (i.e. every orbit is dense)

if and only if X is repetitive. Another fact is the correspondence between unique ergodicity

and the existence of patch frequencies. We recall that a finite Borel measure on ΩX is

translation invariant (or simply invariant) if the measure of every Borel set and the

measure of any of its translations coincide. Let X be a Delone set and p be a patch of X. For

each N > 0 and x ∈ Rd let CN = [−N,N ]d and Lp(x+CN) be the number of patches inside

x+CN that are translated copies of p. A Delone set X has uniform pattern frequencies

if for every patch p the following limit

(1.2.1) freq(p) = limN→+∞

Lp(x+ CN)

vol(x+ CN),

where vol(()x+CN) denotes the d-dimensional volume of x+CN , exists uniformly in x ∈ Rd.

The system (ΩX ,Γ) is called uniquely ergodic if there is a unique invariant probability

measure on it. It is known that the system (ΩX ,Γ) is uniquely ergodic if and only if X has

uniform pattern frequencies (see Lee, Moody and Solomyak [Lee 02] and also Section 2.6).

Linearly repetitive Delone set have uniform pattern frequencies (see Lagarias and Pleasants

[Lagarias 03]). Therefore, the hull ΩX of a linearly repetitive Delone set is minimal and

12

1 Introduction

uniquely ergodic.

1.3 Results of this dissertation

This dissertation consists of two parts: the speed of convergence to patch frequencies for

linearly repetitive Delone sets and a generalization of rotation number to functions that

are equivariant with respect to a Delone set (rather than just periodic), which includes a

generalization of Poincare’s Theorem for circle homeomorphisms. The first part (Chapters 3

and 4) is joint work with Daniel Coronel. The second part (Chapter 5 and 6) will appear in

[Aliste-Prieto 09].

1.3.1 Fast convergence to patch frequencies for linearly repetitive

Delone sets

In the first part of this dissertation we are interested in computing the rate of convergence to

the frequency of a given patch in a repetitive Delone set having uniform pattern frequencies.

Since the results are technical and require some notations that we wish to avoid in this

introduction, we will give here only some motivation and a brief description of the results of

this first part.

We start with the motivation. Let M be a compact metric space, T : M → M a home-

omorphism, and µ an ergodic T -invariant probability measure. Birkhoff’s Ergodic Theorem

states that for every Borel set A the quantity

(1.3.1)1

Ncard

k ∈ 0, . . . , N − 1 : T k(x) ∈ A

→ µ(A) a.e.

when N → +∞. More generally, for every integrable function f : M → R we have

(1.3.2)1

N

N−1∑k=0

f(T k(x))→∫fdµ a.e.

when N → +∞. It is natural to study the rate of convergence in both limits. This is the

subject of several works (see e.g [Petersen 89] and the references therein). It is known that in

general no such rate exists, since there are functions for which the convergence is as slow as

13

1 Introduction

desired (see [Halasz 76]). Thus, to get an answer to the previous question, we need to restrict

to a subclass of functions. In fact, the fastest convergence rate is obtained by supposing that

f is a L∞-coboundary of the system, that is to say, there exists g in L∞(M) satisfying

g T − g = f.

In this case there exists a constant C > 0 such that∣∣∣∣∣ 1

N

N−1∑k=0

f(T k(x))−∫fdµ

∣∣∣∣∣ < C

Na.e.

Moreover, the above convergence rate is attained only by L∞-coboundaries (see [Kachurovskiı 96][Theorem

19]). In the context of (1.3.1), this means that the fastest convergence rate is attained if

µ(A) is a measurable eigenvalue of (M,µ, T ) (see [Halasz 76]).

Now suppose that X is an aperiodic linearly repetitive Delone set. Hence by a result of

Lagarias and Pleasants [Lagarias 03], X has uniform pattern frequencies and (1.2.1) can be

seen as a higher-dimensional strengthened version of (1.3.1). If one refers to the center of

the ball defining an R-patch p as its center , then for a given region D denote by np(D)

the number of R-patches having their centers in D that are translated copies of p. It is not

difficult to check that np(CN)−Lp(CN) = O(Nd−1) and hence freq p can be computed with

np(CN). This is the approach of Lagarias and Pleasants, who actually obtained an upper

bound for the rate of convergence:

Lagarias-Pleasant’s Theorem. Let X be a linearly repetitive Delone set and p ⊂ X be a

patch of X. There exists 0 < δ < 1 such that∣∣∣∣ np(BN)

vol(BN)− freq(p)

∣∣∣∣ = O(N−δ).

We can now raise the main question of the first part of the dissertation:

Question 1. Given a linearly repetitive Delone set X, is there a description of the patches

for which the convergence of np(BN )

vol(BN )to freqp is as fast as possible?

In this dissertation we approach this question by using box decompositions and tower

systems. Box decompositions are a natural generalization of Kakutani-Rohlin partitions

of minimal Cantor systems to the hull of an aperiodic repetitive Delone set, and tower

systems correspond to nested-sequence of Kakutani-Rohlin partitions. They were introduced

14

1 Introduction

by Bellissard, Benedetti and Gambaudo [Bellissard 06]. We give a detailed description of

them in Chapter 3 and we carry out some of the details of the construction to the linearly

repetitive case. This allows us to prove the existence of “good” tower systems (c.f. Theorem

3.5.1).

Then in Chapter 4 we tackle Question 1. We give a sufficient condition (c.f. Theorem

4.1.1) in terms of the “good” sequence of box decompositions defined in Chapter 3 to ensure

fast-convergence. The tools used also allow us to give a proof of Lagarias-Pleasant’s theorem

(c.f. Corollary 4.3.3).

1.3.2 Translation numbers

The second part of this dissertation deals with translation theory for a class of maps of the

real line. In particular, we provide a generalization of the Poincare’s classification for circle

homeomorphisms, which is one of the oldest and most important results in the theory of

dynamical systems. We deviate a little from the standard notation in rotation theory of the

circle in order to simplify the comparison between the Poincare’s classification and our work.

Let f : R → R be a continuous non-decreasing map of the real line. The displacement

of f is the continuous function φ defined by t → f(t) − t. The translation number of f

at t is defined by

(1.3.3) ρ(f, t) = limn→∞

fn(t)− tn

,

provided the limit exists.

Let us suppose, for the moment, that φ is a periodic function and that f is injective.

It follows that f factors through an orientation-preserving homeomorphism of the circle

F : R/Z→ R/Z. If π : R → R/Z denotes the natural projection of the real line R onto the

circle R/Z, this means that π f = F π.

The first result in this topic proved by Poincare is that the translation number ρ(f, t)

exists for every t ∈ R and is independent of t ∈ R. Therefore, there is a translation number

of f . Moreover, if we take a map that also factors through F , then its translation number

differs from the translation number of f by an integer. This allows us to define the rota-

tion number of F as the number ρ(F ) = ρ(f, t) mod Z for any t ∈ R. Poincare then

15

1 Introduction

proved that rotation numbers may be used to classify the dynamics of orientation-preserving

homeomorphisms of the circle.

Poincare’s Theorem. Let F : R/Z → R/Z be an orientation-preserving homeomorphism.

Then:

(1) The rotation number ρ(F ) is rational if and only if F has a periodic orbit.

(2) If the rotation number ρ(F ) is irrational then F is semiconjugate to Rρ(F ), which means

that there exists a continuous, surjective, orientation-preserving map

H : R/Z→ R/Z such that

H F = Rρ(F ) H,

where Rρ(F ) is the rotation x 7→ x+ ρ(F ) mod Z in R/Z.

(3) If in (2) we assume that F is transitive (there is a dense orbit) then the map H defined

in (2) is a homeomorphism.

Before specifying the class of maps we study, we mention a few of the classic generalizations

of the notion of rotation number for more general systems. Newhouse, Palis and Takens

[Newhouse 83] defined and studied a rotation interval for continuous maps on the circle of

degree 1. This was generalized by Misiurewicz and Ziemian [Misiurewicz 89], who gave and

studied various notions for the rotation set of a homeomorphism of the torus Rk/Zk. This

approach was then generalized by Geller and Misiurewicz [Geller 99].

Let X be an aperiodic repetitive Delone subset of the real line. A function φ : R → R is

called strongly X-equivariant or a short-range potential if there exists S > 0, called

the range of φ, such that

(X − t) ∩ [−S, S] = (X − s) ∩ [−S, S]

implies

φ(t) = φ(s).

We say that a continuous function φ : R→ R is X-equivariant if it is the uniform limit of

a sequence of strongly X-equivariant continuous functions.

The main purpose of this part of the dissertation is to study translation (rotation) theory

for continuous non-decreasing maps whose displacements are X-equivariant.

Our first result states

16

1 Introduction

Result 1. Suppose that f : R → R has an X-equivariant displacement φ(t) = f(t) − t and

that φ is bounded away from zero. If the Delone set X has uniform pattern frequencies, then

for every t ∈ R the translation number ρ(f, t) exists and moreover is constant in t. That is

to say, f has a translation number.

It is well known that a periodic continuous function with period 1 function factors through

a continuous function on the circle. An analogous result in the aperiodic case is the following:

if φ is a X-equivariant function, then it factors through a continuous function Φ : ΩX → R.

This function is defined as the unique function satisfying

(1.3.4) φ(t) = Φ(X − t) for all t ∈ R.

Existence comes from the fact that X-equivariance is equivalent to the uniform continuity of

Φ on the orbit of X. Uniqueness follows from the minimality of ΩX which follows from the fact

that X is repetitive. Moreover, every continuous function Φ on Ω induces an X-equivariant

function t→ Φ(X − t) (see e.g. [Kellendonk 03]).

This correspondence may be extended to maps with X-equivariant displacement. Indeed,

for each continuous map f with X-equivariant displacement φ, there is a continuous map

F : ΩX → ΩX defined by

F (Y ) = Y − Φ(Y ) for all Y ∈ ΩX ,

where Φ is the function defined by (1.3.4). We refer to Φ as the displacement of F . We

check that F is homotopic to the identity and that

(1.3.5) X − f(t) = F (X − t) for all Y ∈ ΩX .

In particular, this implies that every Γ-orbit is F -invariant. Since ΩX is locally the product of

a Cantor set and an interval, it follows that every continuous self-map of ΩX must send each

Γ-orbit to another Γ-orbit. Therefore, it is natural to ask whether every continuous self-map

of ΩX for which every Γ-orbit is also F -invariant, induces by (1.3.5) a map of the real line with

an X-equivariant displacement. When we restrict this question to the orientation-preserving

case we obtain

Result 2 (Theorem 5.3.3). Let F : ΩX → ΩX be continuous and orientation-preserving. If

every Γ-orbit is F -invariant then there exists a continuous function Φ : ΩX → R satisfying

F (Y ) = Y − Φ(Y ) for every Y ∈ ΩX .

17

1 Introduction

In particular, Result 2 implies that F is homotopic to the identity and the map defined

by (1.3.5) has X-equivariant displacement. The proof of this result is divided in two parts.

First one shows that it is sufficient to prove that Φ is bounded on an open subset of ΩX .

Then one uses Baire’s Lemma to conclude (the application of Baire’s Lemma was pointed us

by J.Kwapisz).

Let F : ΩX → ΩX be a continuous map. We assume that F is homotopic to the identity and

therefore has a continuous displacement Φ : ΩX → R. The following definitions are adapted

from Geller and Misiurewicz [Geller 99]. The translation number of F at Y ∈ ΩX is

defined by

ρ(F, Y ) = limn→+∞

1

n

n−1∑k=0

Φ(F k(Y ))

provided the limit exists. The (pointwise) translation set of F , denoted by ρp(F ), is defined

as the set of all translation numbers of F . We observe that if Y = X − t with t ∈ R then

1

n

n−1∑k=0

Φ(F k(X − t)) =fn(t)− t

n.

Result 1 is a consequence of the following more general result describing the translation set

of F :

Result 3. Let F : ΩX → ΩX be a continuous map satisfying the hypotheses of Result 2, and

suppose that X has uniform pattern frequencies. If we denote the displacement of F by Φ,

then exactly one of the following assertions holds:

(1) Φ changes signs: for every Y ∈ ΩX , the translation number of F at Y exists and equals

0 (c.f. Proposition 5.4.2);

(2) Φ does not have zeros: F has a unique translation number, which is different from zero

(c.f. Theorem 5.4.7);

(3) Φ does not change signs but has zeros: There exists ρ ∈ R such that for almost every

Y ∈ ΩX , the translation number of F at Y coincides with ρ (c.f. Theorem 5.4.3).

We discuss some previous results that are related to Results 1 and 3: [Kwapisz 00] proved

results that correspond to Result 1 and 3 when the displacement of f is an almost-periodic

function in the sense of Bohr. [Clark 02] studied rotation numbers for self-maps of solenoids

that are homotopic to the identity, so Result 3 may be seen as a generalization of some

of the results of Clark. In particular, Result 3 (3) already appears in [Clark 02]. Finally,

18

1 Introduction

[Shvetsov 03] studied rotation numbers for continuous-time flows on solenoids and generalized

solenoids arising from self-similar tilings. We observe that his results are true for general-

ized solenoids arising from aperiodic repetitive tilings having uniform pattern frequencies.

However, since not every map considered in this dissertation is a time-t map for a flow, our

results are not consequences of the results in [Shvetsov 03].

1.3.3 Towards a Poincare-like classification

In the following, X is an aperiodic repetitive Delone set having uniform pattern frequencies.

Let F : ΩX → ΩX be a map satisfying case (2) of Result 3. That is, F does not have fixed

points, which means it does not have periodic points either and it has a translation number

ρ(F ). Our objective is to classify the dynamics of F in a Poincare-like way. In particular, we

ask whether there exists a semiconjugacy from F to Γρ(F ), i.e., a continuous and surjective

map H : ΩX → ΩX such that

H(F (Y )) = H(Y )− ρ(F ) for every Y ∈ ΩX .

Motivated by the fact that translation numbers are not necessarily preserved by semiconju-

gacies which are not homotopic to the identity, we require that semiconjugacies be homotopic

to the identity. To simplify notations we call such a semiconjugacy a Γ-semiconjugacy .

This allows us to reduce the problem of finding a semiconjugacy to the problem of finding a

continuous solution to the following cohomological equation:

(1.3.6) Ψ(Y − Φ(Y ))−Ψ(Y ) = ρ(f)− Φ(Y ), for all Y ∈ ΩX

where Φ denotes the displacement of F and Ψ corresponds to the displacement of the desired

Γ-semiconjugacy. Cohomological equations are a very important object in the theory of

dynamical systems and appear in several contexts (see for instance Katok and Hasselblat

[Katok 95], Katok and Robinson [Katok 01]).

A direct necessary condition for the existence of a continuous solution for (1.3.6) is the

following: there exists C > 0 such that

|F n(Y )− Y − nρ(F )| < C for every n ∈ N and Y ∈ ΩX .

19

1 Introduction

In the periodic case, this condition is always satisfied and it is a key ingredient of Poincare’s

Theorem. In our setting, it is not known whether this condition is satisfied by all F ’s, so we

say that a map that satisfies this condition is ρ-bounded . In the following, we suppose that

F is ρ-bounded.

The main result of this part of the dissertation is a generalization of Poincare’s Theorem

parts (1) and (2), where a key role was played by rational numbers. In our case, rational

numbers are no longer available. We also point out that translations in ΩX have no periodic

points since X is aperiodic. To define an alternative set, we observe that if F is minimal

and ρ-bounded then a well-known theorem of Gottschalk and Hedlund ensures the existence

of a continuous solution to the cohomological equation and hence also the existence of a

semiconjugacy from F to Γρ(F ), and this implies that the translation Γρ(F ) is minimal. The

well known fact that a rotation of the circle is minimal if and only if the angle defining it is

not rational motivates us to define

Q = t ∈ R : Y ∈ ΩX 7→ Y − t is not minimal on ΩX.

Before stating our main result we need some definitions, which are inspired by Jager and

Stark [Stark 03, Jager 06]. However, in our setting transverse sets are totally disconnected,

and this forces us to work locally.

Recall that local verticals are clopen subsets of cylinders and hence they are Cantor sets.

Given a local vertical V in ΩX and two functions α, β : V → R, the set

V [α, β] = X − t : X ∈ V, α(X) ≤ t ≤ β(X)

is called a local strip. A local strip V [α, β] is called thin if V [α, α] = V [β, β] = V [α, β]. In

particular, thin strips do not have interior. Finally, we state the main result of this part of

the dissertation:

Result 4. Suppose that F is a ρ-bounded map preserving the orientation in ΩX with trans-

lation number ρ(F ). Then:

• If ρ(F ) ∈ Q, then F is not minimal and every minimal set is the finite disjoint union

of thin local strips (c.f. Theorem 6.3.6).

• If ρ(F ) 6∈ Q, then F is Γ-semiconjugated to Γρ(f) (c.f. Theorem 6.3.8).

20

2 Tilings, Delone sets and Delone

dynamical systems

In this chapter we recall some basic notions about tilings and Delone sets. We follow the

fruitful approach which associates a dynamical system to either a tiling or a Delone set whose

dynamical properties reflect some combinatorial properties of the given object. For a more

introductory exposition, we refer the reader to [Lagarias 03, Robinson 04, Solomyak 97].

2.1 Delone sets

For d ∈ N fixed, let Rd denote the d-dimensional Euclidean space endowed with the usual

Euclidean norm, which is denoted by ‖ · ‖. For x ∈ Rd and S > 0, the open (respectively

closed) ball around x of radius S will be denoted by BS(x)(respectively BS(x)).

A discrete set X ⊆ Rd is called a Delone set if it satisfies the following properties:

• Uniformly discrete : there exists r > 0 such that every closed ball of radius r meets

X in at most one point;

• Relatively dense : there exists R > 0 such that every closed ball of radius R meets

X in at least one point.

Given a Delone set X, we define its proximity radius as

(2.1.1) r(X) =1

2inf‖x− y‖ |x, y ∈ X, x 6= y,

and its occurrence radius as

(2.1.2) R(X) = infR > 0 : X ∩BR(y) 6= ∅ for all y ∈ Rd.

21

2 Tilings, Delone sets and Delone dynamical systems

We observe that r(X) is the supremum of the constants r > 0 such that X satisfies the

“Uniformly discrete” property and R(X) is the infimum of the constants R > 0 such that X

satisfies the “Relatively dense” property and we took the two definitions from [Besbes 08].

The group of rigid translations on Rd will be denoted by Γ. More precisely Γ = Γv : v ∈Rd where Γv : Rd → Rd is defined by Γv(x) = x−v. Given a Delone set X ⊆ Rd and v ∈ Rd,

the set X − v is the Delone set defined by translating each point of X by v, namely,

X − v = x− v : x ∈ X.

A Delone set X ⊆ Rd is of finite type if

X −X = x− y : x, y ∈ X

is locally finite, i.e., its intersection with every bounded region is finite.

The period lattice of a Delone set X in Rd is the lattice of translation symmetries ΛX of

X, given by

ΛX = v ∈ Rd|X − v = X.

It is a free abelian group with rank smaller than or equal to d. The most regular Delone sets

are those with a full set of translation symmetries, i.e., rank(ΛX) = d and we call such an X

an ideal crystal . We call X aperiodic if rank(ΛX) = 0, which means that ΛX = 0.

For R > 0 and y ∈ Rd, the subset p ⊆ X defined by p = X ∩By(R) is called the R-patch

of X centered at y ∈ Rd. When there is no confusion, we will refer to p simply as a patch.

A patch q is a translated of the patch p if there exists v ∈ Rd such that p− v = q.

Suppose that p = X∩BR(x) is a patch of X. The vector v ∈ Rd is called a return vector

of p in X if p + v is a patch of X. The set of all return vectors of p in X is denoted by

Rp(X). Notice that for every v ∈ Rp(X) we have

Rp(X) = Rp+v(X)− v.

The Delone set X is repetitive if for each R > 0 there is a finite number M > 0, such

that for every closed ball B of radius M the set B ∩X contains a translated patch of every

R-patch of X. Equivalently, the set Rp(X) of return vectors of p in X is a Delone set for

22


every R-patch p of X and every R > 0. Observe that any repetitive Delone set is necessarily

of finite type.

The R-atlas AX(R) of X is the collection of all the R-patches centered at a point of X

translated to the origin. More precisely:

AX(R) = X ∩BR(x)− x : x ∈ X.

The atlas AX of X is the union of all the R-atlases, for R > 0. Notice that X is of finite

type if and only if AX(R) is finite for every R > 0.

If X is repetitive then for each R > 0 the smallest M > 0 such that the ball of radius M

contains a translated of every patch in AX(R) is denoted by MX(R). The function MX is

called the repetitivity function of X. The Delone set X is called linearly repetitive if

there exists a constant L > 0 such that every ball of radius LR contains a translated of every

possible R-patch for every R > 0. Written in terms of the repetitivity function, a Delone set

is linearly repetitive if and only if

(2.1.3) MX(R) ≤ LR for all R > 0.

2.2 Delone systems

Denote by D the collection of all the Delone sets of Rd . We endow D with the so-called

tiling topology , which is induced by the following metric (see [Robinson 04] for details):

given X1, X2 two Delone sets in D, define

d(X1, X2) = mind(X1, X2), 2−1/2

where

d(X1, X2) = infε > 0 | ∃x ∈ Bε(0), (X1 − x) ∩B1/ε(0) = X2 ∩B1/ε(0).

This means that two Delone sets in D are close if, after a small translation, they coincide in

a big ball around the origin. It is easy to check that rigid translations are continuous with

23


respect to the tiling topology. Moreover, the action of Γ over D defined by

(x,X) 7→ X − x

is continuous in both variables. By an abuse of notation, we will identify Γx with the

homeomorphism given by X 7→ X − x, i.e., we denote ΓxX = X − x. Given a Delone set X

in D, its orbit is defined by

OΓ(X) := X − x : x ∈ Rd.

A Γ-invariant set is a set K ⊆ D such the orbit of every X in K is contained in K. Thus,

Γ-invariant sets are the union of their orbits.

Disgression: We check that ΛX is the stabilizer group of X under the action Γ and thus

OΓ(X) can be identified to Rd/ΛX , which is then equipped with two topologies: the natural

quotient topology and the tiling topology we just have defined. In general the first one is

coarser than the second one, except in the case of an ideal crystal where these two topologies

coincide and OΓ(X) is homeomorphic to a d-dimensional torus. In the aperiodic case, OΓ(X)

is identified with Rd but the tiling topology is strictly finer than the Euclidean topology and,

in particular, OΓ(X) is not complete with respect to the tiling topology.

A Delone system is a pair (Ω,Γ) where Ω is a Γ-invariant closed subset of D. Given

X ∈ D we define its continuous hull as

ΩX := OΓ(X).

It is clear that (ΩX ,Γ) is a Delone system. When X is of finite type, then ΩX is compact

(see [Rudolph 89, Robinson 04]).

A Delone system (Ω,Γ) is called minimal if the empty set is the only closed proper Γ-

invariant subset of Ω. Equivalently, (Ω,Γ) is minimal if and only if Ω = ΩX for every X ∈ Ω,

i.e., Ω is the union of its orbits and each orbit is dense in Ω.

The minimality of the hull of a Delone set of finite type can be characterized in combina-

torial terms as follows:

Theorem 2.2.1 ([Lagarias 03, Theorem 3.2]). Let X be a Delone set of finite type. Then

the Delone system (ΩX ,Γ) is minimal if and only if X is repetitive. In such case, one also

24


has

ΩX = Y ∈ D |AY = AX.

As the occurrence and proximity radii of a Delone set depend only on its atlas of patches,

the previous theorem yields:

Corollary 2.2.2. For a minimal Delone system Ω, the occurrence and proximity radii defined

in (2.1.1) and (2.1.2) are constant in Ω.

A Delone system is called aperiodic if every Delone set in it is aperiodic. We observe

that if ΩX is the hull of an aperiodic Delone set X then it is not necessarily true that ΩX is

aperiodic. To see this, it suffices to consider the continuous hull of X = Zd \ 0. Indeed,

the integer lattice Zd belongs to ΩX since d(X− (k, 0, . . . , 0),Zd) converges to 0 as k → +∞.

However, this example is not minimal. In the minimal case, one do has the following result:

Proposition 2.2.3 ([Kellendonk 00]). Let X be a repetitive Delone set. Then (ΩX ,Γ) is

aperiodic if and only if X is aperiodic.

2.3 Tilings

A tile in Rd is a compact subset of Rd that is homeomorphic to a closed ball. A tiling Tof Rd is a countable cover T = t1, t2, . . . , of Rd by tiles having pairwise disjoint interiors.

If every tile of a given tiling is a polyhedron1 then the tiling is said to be polyhedral 2. A

Penrose tiling is an example of a polygonal tiling (see Figure 1.4). A patch p of a tiling T is

a finite subset of it. Its support , denoted supp(p), is defined as the union of its tiles and its

diameter is defined as the diameter of its support. The group of rigid translations Γ acts

over any set of tiles the same way it acts over Delone sets. More precisely, if v ∈ Rd and A

is a set of tiles, then

ΓvA = A− v := a− v : a ∈ A.

One says that a tiling has finite local complexity if for every R > 0 the number of

patches of diameter less than R is finite modulo translations. A tiling T is repetitive if for

1polygon if d = 22polygonal if d = 2

25


every patch p there is R > 0 such that every ball of radius R contains a translated copy of

p. Equivalently, a tiling T is repetitive if for every patch p the set of return vectors

Rp(T ) = x ∈ Rd |p + x ⊆ T

is a Delone set.

It will be convenient to consider decorated tilings. A punctured tile is a pair (t, y) where

t is a tile and y is a point in the interior of t. One refers to y as the puncture of t. A

punctured tiling of Rd is a countable collection (ti, yi)i of punctured tiles such that tiiis a tiling of Rd. Translations induce an equivalence relation over tiles: two tiles t, t′ are

equivalent if there exists x ∈ Rd such that t − x = t′. If the tiles are punctured, then it is

also required that the translation by v sends the puncture y of t to the puncture y′ of t′, i.e.,

y − v = y. Equivalence classes of (punctured) tiles are called prototiles .

Figure 2.1: A punctured Penrose tiling

We will also consider colored tilings (possibly punctured), that is, we give to each tile t

a color from a finite predefined set of colors. Two colored tiles (resp. colored punctured tiles)

are equivalent if they are equivalent as tiles (resp. as punctured tiles) and they have the same

color. The definitions of tiling systems are analogous to the definitions of Delone systems.

The term decorated tiling is used to refer to a tiling that is colored and/or punctured.

26


2.3.1 From tilings to punctured tilings and Delone sets

Let P = p1, . . . , pn be a finite set of (possibly colored) prototiles. Consider a tiling T with

prototiles in P . Choosing a point yi in the interior of a tile tpi of each prototile pi naturally

defines a finite set of punctured prototiles. Namely, if a tile t belongs to pi then there is

v ∈ Rd such that tpi − v = t and thus yt = (t, yi − v) is a puncture of t that makes it

equivalent to (tpi, yi).

Consider a tiling T with prototiles in P . We carry out the former process for each tile in

T and define in this way a punctured tiling Tpunc. The set of punctures of Tpunc is defined

as

XT = yt : t ∈ T .

Since P is finite, it follows that XT is a Delone set. The following result relates properties

of XT to properties of T .

Proposition 2.3.1. Let T be a tiling with prototiles in P. Suppose that P is punctured as

explained above and consider the punctures’ set XT of T . Then the following assertions hold:

• T has finite local complexity if and only if XT is of finite type.

• T is repetitive if and only if XT is repetitive.

2.3.2 Voronoi tilings: from Delone sets to tilings

In this section we describe a standard method to derive a tiling starting from a Delone set,

by using Voronoi cells. Given a Delone set X, the Voronoi cell Vx(X) of a point x in X is

defined by:

Vx(X) = y ∈ Rd | ∀x′ ∈ X, ||y − x|| ≤ ||y − x′||.

It is well known that Voronoi cells are closed convex subsets of Rd (polyhedra if X is of finite

type) and that two different Voronoi cells may intersect only at their boundaries. Moreover,

they allow us to define the following notion of proximity in X: two points x, y ∈ X are

neighbors (in X) if their Voronoi cells intersect in at least one point.

We recall that the occurrence radius of X is defined by

R(X) = infR > 0 : X ∩BR(y) 6= ∅ for all y ∈ Rd.

27


The next lemma states that the sizes of Voronoi cells of X are controlled by the occurrence

radius of X (see e.g. [Besbes 08] for a proof).

Lemma 2.3.2. Let X be a Delone set in Rd and suppose that R > R(X). Then, for every

x ∈ X, the Voronoi cell Vx(X) around x is included in the ball of radius R around x.

Furthermore, the ball B2R(x) contains all the neighbors of x.

We define the Voronoi tiling TX of the Delone set X as the collection of all Voronoi cells

of X. That is, TX is defined by

TX = Vx(X) : x ∈ X.

We remark that the fact that TX is a tiling of Rd is direct from the definition of Voronoi cell.

Moreover, the tiles in TX are polyhedra and meet full-face to full-face, i.e., the intersection

of any two tiles is either empty or a face (of every dimension smaller than d). In Figure 2.2,

we see a region of the Voronoi tiling of the Delone set that arises by considering the set of

punctures of the punctured Penrose tiling that appears in Figure 2.1.

Figure 2.2: A region of Voronoi tiling associated to the set of punctures of a puncturedPenrose tiling

The following result gives the basic correspondence between Delone sets and their Voronoi

tilings. Its proof, which is omitted, follows easily from Lemma 2.3.2.

28


Proposition 2.3.3. Let X be a Delone set. Then the following assertions are true:

• X is of finite type if and only if TX has finite local complexity;

• X is repetitive if and only if TX is repetitive.

2.4 The canonical transversal

Let X ⊆ Rd be a Delone set of finite type and ΩX its continuous hull. The canonical

transversal of ΩX is defined as the collection of all Delone sets in ΩX that contain the

origin, i.e.,

Ω0X = Y ∈ ΩX | 0 ∈ Y .

The canonical transversal is clearly closed. We remark that for every Y ∈ ΩX , a point x ∈ Rd

belongs to Y if and only if Y − x ∈ Ω0X . Besides, by Corollary 2.2.2 all Delone sets in ΩX

have the same proximity radii r(X). This means that a ball of radius r(X) around 0 of every

Delone set Y in Ω0X does not contain other points of Y , which by the previous remark implies

that Y − v 6∈ Ω0X for every v ∈ Br(X)(0). It follows that Ω0

X is totally disconnected.

Given a R-patch p centered at x0 ∈ Rd, the cylinder of p consists of all Delone sets whose

R-patch centered at 0 is equivalent to p. Namely,

Cp = Y ∈ ΩX : Y ∩BR(0) = p− x0.

Given a Delone set Y ∈ ΩX and R > 0, the cylinder around Y of range R is defined as

CY,R := W ∈ ΩX : W ∩BR(0) = Y ∩BR(0),

i.e., CY,R = CY ∩BR(0). It is easy to see that given two Delone sets Y,W and R > 0, the

cylinder sets CY,R and CW,R either coincide or they are disjoint. Moreover, if Uε(Y ) denotes

the ball around Y ∈ Ω0X of radius ε in ΩX then

Uε(Y ) ∩ Ω0X = CY,1/ε.

This implies that a sequence (Yn)n∈N ⊆ Ω0X converges to some Y ∈ Ω0

X if and only if for

every ε > 0 there exists n0 ∈ N such that

Yn ∩B1/ε(0) = Y ∩B1/ε(0)

29


for every n ≥ n0. It also implies that the cylinders CY,R are clopen (both closed and open)

subsets of Ω0X when Y ∈ Ω0

X and they form a basis of Ω0X . Hence we have

Proposition 2.4.1 ([Kellendonk 00]). Let X be a Delone set of finite type. Then Ω0X is

closed and totally disconnected. If X is aperiodic and repetitive then Ω0X is a Cantor set.

2.5 Entrance and return times

Let X be an aperiodic repetitive Delone set and ΩX its hull. Let U be a subset of ΩX . The

set of entrance times of a Delone set Y ∈ ΩX to U is defined as

(2.5.1) RU(Y ) = x ∈ Rd |Y − x ∈ U.

When the Delone set Y belongs to U , the set RU(Y ) is called the set of return times of

Y to U .

A clopen subset C of a cylinder is called a local vertical . As the following result shows,

the set of return times to a local vertical has some remarkable properties that will be used

throughout this dissertation.

Lemma 2.5.1. Let C be a local vertical in ΩX . Then, for every Y ∈ C the set of return

times RC(Y ) is a repetitive Delone set whose patches depend only on C. In particular, the

occurrence and proximity radii of RC(Y ) depend only on C.

Proof. By definition of local vertical there exists R > 0 and an R-patch p of X (and therefore

of Y ) such that C ⊆ Cp. It follows that RC(Y ) ⊆ RCp(Y ) = Rp(Y ). The set Rp(Y )

is a Delone set since Y is repetitive by minimality of (ΩX ,Γ). This clearly implies that

RC(Y ) is uniformly discrete. To see that RC(Y ) is relatively dense, consider the open set

Z − t : Z ∈ C, t ∈ (−ε, ε) with epsilon small enough.

The minimality of ΩX implies that Y is repetitive, which means that The relative density

of RC(Y ) follows easily from the minimality of ΩX .

We check now that RC(Y ) is repetitive. Indeed, fix r > 0 and let p = RC(Y ) ∩ Br(x) be

the r-patch of RC(Y ) around x ∈ RC(Y ). Thus Y − x is in C. Since C is local vertical,

there is s > 0 such that V = CY−x,s where s is big enough so that every Y ′ in V agree with

30


Y − x in the closed ball of radius r around 0. Since RV (Y − x) is relatively dense there

exists M > 0 such that every closed ball of radius M in Rd intersects RV (Y − x). Since

RV (Y − x) + x ⊆ RC(Y ) every closed ball of radius M + r contains a translated of p.

Finally we show that RC(Y ) and RC(Y ′) have the same atlases of patches. Let p be an

r-patch of RC(Y ) centered at x ∈ Rd. Consider the cylinder Cp−x in C of all Delone sets

whose r-patch centered at 0 is p − x. By minimality there exists x′ in RC(Y ′) such that

Y ′−x′ is in Cp which implies that the r-patch B(x′, r)∩RC(Y ′) is a translated copy of p.

Remark 2.5.2. Thanks to the previous lemma, we write R(C) = R(RC(Q)) and r(C) =

r(RC(Q)) for the occurrence and proximity radii of RC(Q) of any Q ∈ C.

2.6 Unique ergodicity versus uniform pattern frequencies

The material in this section follows from [Lee 02, Lagarias 03]. Given a bounded set D ⊂ Rd,

let

D+r = x ∈ Rd | d(x,D) ≤ r.

A sequence (Dn)n∈N of bounded measurable subsets of Rd is called a Van Hove sequence

if for every r > 0,

limn→+∞

vol((∂Dn)+r)

vol(Dn)= 0,

where ∂D denotes the boundary of a given set D.

Let p be a patch in X and D ⊂ Rd. Define Lp(D) to be the number of patches of X

included in D that are equivalent to p.

The Delone set X is said to have uniform pattern frequencies (with respect to

(Dn)n∈N) if for every patch p, the limit

freq(p) = limn→+∞

Lp(Dn + x)

vol(Dn)

exists uniformly in x ∈ Rd.

A finite Borel measure µ on ΩX is called Γ-invariant (or simply invariant) if for every

31


measurable set B it satisfies

µ(ΓxB) = µ(B), for every x ∈ Rd.

We denote the set of Γ-invariant probability measures on ΩX by MI(ΩX). By a standard

argument, the set MI(ΩX) is not empty and ΩX is called uniquely ergodic whenever

MI(ΩX) contains a unique element. The following result is the uniquely ergodic version of

Birkhoff’s Ergodic Theorem for Rd-actions:

Theorem 2.6.1 ([Lee 02, Theorem 2.6]). Let X be a Delone set of finite type and (Dn)n∈N

a Van Hove sequence. Then (ΩX ,Γ) is uniquely ergodic if and only if for every continuous

function f : ΩX → C the limit

c = limn→+∞

1

vol(Dn)

∫Dn

f(Y − x)dx

exists uniformly in Y ∈ ΩX and does not depend on Y ∈ ΩX . Moreover, in such case

c =∫

ΩXfdµ, where µ is the unique Γ-invariant probability measure on ΩX .

It is clear that if ΩX is uniquely ergodic then X has uniform pattern frequencies. Further-

more, the converse is also true for Delone sets of finite type:

Theorem 2.6.2 ([Lee 02, Theorem 2.7]). Let X be a Delone set of finite type. Then (ΩX ,Γ)

is uniquely ergodic if and only if X has uniform pattern frequencies (with respect to any Van

Hove sequence).

Instead of counting complete occurrences of patches inside a region we could choose to

count the centers of translated copies of a given patch inside a region. Thus, np(Dn) will

be the number of copies of p having the center inside Dn. It is not so difficult to see that

np(Dn)/ vol(Dn) will converge to the freq(pn) if and only if Lp(Dn)/Dn does. However, np

has some additivity properties that will be used in Chapter 4 to give a new proof of the

following result:

Theorem 2.6.3 ([Lagarias 03, Theorem 6.1]). Let X be a linearly repetitive Delone set.

There is 0 < δ < 1, which depends only on X, such that for every r-patch p of X and every

N > 0, there are a number freq(p) > 0 and a constant C > 0(that may depend on p) such

that

|np(BN)−Nd freq(p)| < CNd−δ

where BN = [−N/2, N/2]d.

32


Since one can easily verify that (BN)N∈N is a Van Hove sequence, the previous results

imply, in virtue of Theorem 2.6.2, that the continuous hull of a linearly repetitive Delone set

is uniquely ergodic.

2.7 Laminations and Rd-solenoids

Laminations or Foliated Spaces are very important objects in geometry and dynamics. We

limit our exposition only to the definitions and results we will need in the sequel. We mainly

follow [Bellissard 06, Benedetti 03] and the interested reader can see [Ghys 99, Moore 06] for

more general references.

Let M be a compact metric space. A foliated chart is a pair (U, h) where U is an open

set in M and h : U → D × T is a homeomorphism where D is an open set in Rd and T is

a separable topological space. The space T is called a transversal space and U is said to

be a box . A slice is a set of the form h−1(D × t) with t ∈ T . One verifies that U is the

union of its slices and that each slice has a natural structure of a d-manifold. We say that

U is foliated by its slices. An atlas for the lamination structure is a collection of foliated

charts (Ui, hi)i such that the transition maps hi,j = hj h−1i satisfy, over their domain

of definition,

(2.7.1) hi,j(v, t) = (fi,j(v, t), γi,j(t))

where γi,j is continuous and fi,j(·, t) is C∞ for every t and all the partial derivatives are

continuous in t. Two atlases (for the lamination structure) are called equivalent if their

union is also an atlas of a lamination structure. The pair (M,L) where L is an equivalence

class of atlases (for the lamination structure) is called a lamination .

One sees from (2.7.1) that each slice can intersect at most one slice of another chart.

This allows to define the leaves of L as the smallest connected subsets that contain all the

slices they intersect. It follows also that leaves inherit a d-manifold structure. We say that

(M,L) is minimal if all its leaves are dense in M . The lamination is orientable if fi,j(·, t)is orientation-preserving for every t.

The trivial example of a lamination is given by a d-manifold where all the transversal

spaces are singletons. A less trivial example is given by foliations of d-manifolds where the

transversals spaces are homeomorphic to Rk for some k ∈ N. A more interesting example,

33


for our purposes, is the following:

Example 2.1. Let (M,T ) be a minimal Cantor system, i.e., the set M is a Cantor set and

T : M → M is a homeomorphism such that the T -orbit of every point is dense in M . Let

f : M → R be a continuous function with f(m) > 0 for all m ∈ M . The special flow of

(M,T ) under f is defined as follows: Set Mf = M × R. There is a natural continuous flow

on Mf : (m, t)τ→ (m, t + τ). Define ∼ to be the smallest equivalence relation on Mf such

that (m, f(m)) ∼ (T (m), 0) and consider the set Mf = Mf/ ∼ of equivalence classes in Mf .

The flow on Mf commutes with the canonical projection π : Mf → Mf and hence we can

define a continuous flow Ttt∈R over Mf . The system (Mf , Ttt∈R) is the special flow of

(M,T ) under f and it is a less trivial example of a lamination where the transversals spaces

are Cantor sets.

In the previous example, the foliated charts not only define an atlas for a lamination

structure, but their transition maps also satisfy the stronger property:

(2.7.2) hi,j(x, t) = (x− gi,j, γi,j(t))

where gi,j ∈ Rd depends only on the charts (Ui, hi) and (Uj, hj) and the map γi,j is continuous.

As we will see in the following section, Delone systems (and tiling systems) have a natural

lamination atlas whose transition maps also satisfy (2.7.2). This key observation motivates

the following definition first given in [Bellissard 06] for Rd-actions and then generalized in

[Benedetti 03] for more general Lie group actions.

Given a compact metric space M , an atlas for the structure of Rd-solenoid is a collection

of foliated charts (Ui, hi : Ui → Di × Ti)i where the transversals spaces Ti are totally

disconnected and the transition maps hi,j satisfy (2.7.2) in their domains of definition. Two

atlases for the Rd-solenoidal structure are called equivalent (for the Rd-solenoidal structure)

if their union is also an atlas (for the Rd-solenoidal structure). A Rd-solenoid is a pair

(M,F) where M is a compact metric space and F is an equivalence class of atlases (for the

Rd-solenoidal structure) and every leaf is M diffeomorphic to Rd.

Since (2.7.2) implies (2.7.1), every atlas for the Rd-solenoidal structure is also an atlas for

the lamination structure. Moreover, two atlases that are equivalent for the Rd-solenoidal

structure are also equivalent for the lamination structure. It follows that if F and L are

the equivalence classes of a given atlas with respect to the Rd-solenoidal and the lamination

structure respectively then F ⊆ L. In this case we write L(F) := L and refer to L(F) as

34


the natural lamination structure associated to F . The leaves of (M,F) are simply the leaves

of (M,L). The Rd-solenoid (M,F) is minimal if and only if (M,L) is minimal, i.e., all

the leaves are dense in M . The rigidity in the transition maps for a Rd-solenoid gives the

following additional structure on leaves:

Proposition 2.7.1 ([Bellissard 06]). Let (M,F) be a Rd-solenoid. Then:

• Every leaf of (M,F) carries a natural flat Riemannian metric. In particular, every leaf

is isometric to Rd.

• There is a free continuous action Γ of Rd over M such that orbits of Γ coincide with

leaves of (M,F).

Proof. We give an outline of the proof. For the first part, given a foliated chart (Ui, hi) one

can push-forward the Euclidean Riemannian metric on Di by the map hi to every slice in Ui.

By (2.7.2), it follows that the Riemannian metric on each slice does not depend on the chart.

Therefore, this procedure defines a Riemannian metric in every leaf and it is clear that each

leaf is isometric to Rd.

For the second part, one can define a “local” action as follows: Take x ∈M and let (Ui, hi)

be a foliated chart such that x ∈ Ui. Set (d, t) = hi(x). If v ∈ Rd is small enough (in norm)

then d− v ∈ Di and we define Γvx = h−1(d− v, t). By virtue of (2.7.2), Γvx is well-defined

and does not depend on the chart (Ui, hi). It is standard to extend this to a global action

and it is easy to check that the action defined in this way is indeed continuous and the orbits

coincide with leaves.

2.8 Delone systems are Rd-solenoids

In this section, we prove that an aperiodic minimal Delone system3 Ω has a natural Rd-

solenoidal structure. This property first appeared in [Bellissard 06] and we give the proof

here for completeness. We will split it into two steps:

First step: Ω is locally homeomorphic to the product of an open set in Rd by a Cantor set.

For every X ∈ Ω and ε > 0 to be chosen let U = Uε(X) be the open ball of radius ε around

3this applies for tiling systems as well.

35


X in Ω and D = z ∈ Rd : ‖z‖ < ε be the open ball of radius ε around z in Rd. By the

definition of the metric, for every Y ∈ U there is y ∈ D such that Y − y coincides with X

on the ball of radius 1/ε around the origin. Choosing ε < minr(X)/2, 1/(R(X) + r(X))where r(X) and R(X) are defined in (2.1.1) and (2.1.2) respectively, ensures that y is uniquely

determined. Indeed, suppose there is y′ ∈ D such that Y − y′ also coincides with X on the

ball of radius 1/ε around the origin. By the choice of ε, there exists x ∈ BR(0)∩X such that

Bε(x) ∩ X = x. It follows that x ∈ Y − y which implies that x + y − y′ ∈ Y − y′. But

‖x+ y− y′‖ < R+ r/2 + r/2 < 1/ε and ‖y− y′‖ < ε. The fact that Y − y′ coincides with X

in B1/ε(0) implies that x+ y − y′ ∈ X. But then x+ y − y′ = x and y = y′. Hence, letting

T (X) = CX,1/ε = Z ∈ Ω : Z ∩B1/ε(0) = X ∩B1/ε(0)

implies that the map h : U → D × T given by the formula

h(Y ) = (z, Y − z), Y ∈ U

is a homeomorphism.

Second step: The local homeomorphisms from the previous step define foliated charts for

Ω covering Ω. We check that the associated transition maps are compatible with the Rd-

solenoidal structure. Since the charts cover the space, by compactness we let (Uk, hk)k to

be a finite subcover of foliated charts. Fix i, j and suppose that Y ∈ Ui∩Uj with Ui = U(Xi)

and Uj = U(Xj). Let hi(Y ) = (yi, Y − yi) and hj(Y ) = (yj, Y − yj). It follows that

(2.8.1) hi,j(yi, Y − zi) = (yi − (yi − yj), Y − yi + (yi − yj)),

Thus, to see that these charts define an atlas for the Rd-solenoidal structure it suffices to

show that vi,j := yi − yj does not depend on Y ∈ Ui ∩Uj and that the map Y − zi 7→ Y − zjis continuous. The former follows from the same argument we use to prove the uniqueness

of y in the previous step because yj − yi is small and Xi coincides with Xj + yj − yi in a

big enough ball around 0. The latter follows from the fact that if Znn ⊆ Ti converges to

Y − yi then there is Sn → +∞ as n→ +∞ such that for every n ∈ N, Zn coincides with Y

in the ball of radius Sn around 0. Indeed, this implies that for n big enough the Delone set

Zn + vi,j coincides with Y − yj in the ball of radius Sn − ‖vi,j‖ > 0 around 0. From this one

checks that Zn + vi,j ∈ Tj and Zn + vi,j → Y − yj as n→ +∞.

36


Figure 2.3: yi − yj does not depend on Y

2.8.1 Local verticals and local transversals

Let (Ω,Γ) be a minimal Delone system and consider F its natural atlas for the Rd-solenoidal

structure(as defined above). A transversal for Ω is a compact subset Ξ of Ω that intersects

every leaf of M in a set that is discrete with respect to the standard topology of the leaf (that

is to say, the topology induced on the leaf by identifying it with Rd). Thus, the canonical

transversal is a transversal in the sense we just defined.

Recall that a local vertical is a clopen subset of a cylinder CY,R with Y in Ω and R > 0.

In terms of the atlas F , the intersection of a local vertical V with every box B with foliated

chart h : B → D × T satisfies h(B ∩ V ) = F × VT , where F is a finite subset of D and VT

is a clopen subset of T . Given a local vertical V in Ω and a function ϕ : V → Rd, we define

the set

V [ϕ] = X − ϕ(X) |X ∈ V .

When the induced map X ∈ V 7→ X−ϕ(X) is bijective we say that V [ϕ] is a local graph . If

in such case the function ϕ is also continuous, then we say that V [ϕ] is a local transversal .

It is standard that local transversals are Cantor sets with the induced topology from Ω (just

follow the same outline given for cylinders in Section 2.4).

37


2.8.2 Invariant measures and transverse invariant measures

In this subsection we will give the definition of transverse invariant measures on the Delone

system Ω and relate this to Γ-invariant measures on Ω. This relationship exists on more

general laminations as well, for which we refer to [Ghys 99]. We also point out that when

d = 1 the transverse invariant measures are the measures induced by the invariant measures

of the flow over the Poincare map on a cross-section, which is isomorphic to a special flow

(see [Cornfeld 82] for details).

A finite transverse invariant measure on (Ω,F) is defined by a collection µii of

finite positive measures (each µi is defined on the transversal space Ti) such that for every i

and j we have

(2.8.2) µi(B) = µj(γij(B))

for every Borelian B in Ti that is contained in the domain of the transition map γij.

Let µ be a finite Γ-invariant measure on Ω. This invariant measure defines a transverse

invariant measure µt of the lamination (Ω,F) as follows: denote by λd the Lebesgue measure

in Rd and for a Borelian subset B of Ti, define

µi(B) = limr→0+

1

λd(Br(v)µ(h−1

i (Br(v)×B)),

where v ∈ Di.

It can be checked that the collection µi defines a transverse invariant measure. In

particular, this formula can be applied to every local vertical and it follows that if h : B →D × Cp is a foliated chart with D ⊆ Rd and p a patch of a Delone set X then

µ(B) = µt(Cp)λd(D),

so we may rewrite Corollary 2.8 in [Lee 02] as:

Proposition 2.8.1. Let X be a Delone set in Ω and suppose that (Ω,Γ) is uniquely ergodic.

Then for every patch p, we have

freqX(p) = µt(Cp)

where µt is the transverse invariant measure induced by the unique invariant measure on Ω.

38

3 Hierarchy of aperiodic Delone sets

In this chapter we introduce tower systems for the hull ΩX of an aperiodic repetitive Delone

set X, which are generalizations of the well-known Kakutani-Rohlin towers for minimal

Cantor systems. In particular, they provide a powerful tool to tackle questions about the

dynamics of the flow Γ. We introduce box decompositions and their derived tilings in Section

3.1. Voronoi box decompositions are then introduced in Section 3.3. Later, in Section 3.4,

following [Bellissard 06], we introduce tower systems and prove a lemma which allows to

show the existence of tower systems. Finally, in Section 3.5 we show the existence of a tower

system for ΩX satisfying some additional “good” conditions when X is linearly repetitive.

3.1 Boxes and canonical parametrizations

Recall that ΩX possesses a natural structure of Rd-solenoid (see Section 2.8). In this context,

leaves coincide with Γ-orbits and boxes are the distinguished open subsets of ΩX that are the

domain of a foliated chart in the maximal atlas (for the Rd-solenoidal structure) F of ΩX .

Alternatively, a box in ΩX is any set of the form

B = Γ(D × T ) = Y − v : Y ∈ T, v ∈ D,

where D is an open subset of Rd, T is a local vertical in ΩX and the restriction of the action

Γ to D × T is one-to-one. The pair (D × T,Γ|D×T ) is called a parametrization of B.

Moreover, if h is the foliated chart associated to B, then h−1 = Γ|D×T . This means that

each set of the form Γ(D × Y ) with Y ∈ T is a slice in B, and each set of the form

Γ(v× T ) = T − v with v ∈ D is a local vertical in B. It turns out that, up to the choice

of a local vertical, there is a canonical parametrization of B, which is defined as follows. Let

C be a vertical in B. Since C = T − v with v ∈ D, it is clear that (D − v × C,Γ) is a

parametrization of B, for which 0 ∈ D− v and T − v ⊂ B. Thus, when parametrizing a box

39


B by D × C we will always assume that C ⊂ B and that 0 ∈ D. We call C the base of B,

D the height of B, and we say that B is a box with base . To simplify the notation we will

identify B and D × C so we write B = D × C instead of B = Γ(D × C). In this chapter we

will reduce to work with boxes for which the slices are homeomorphic to an open ball in Rd.

Now we introduce two notions that allow us to describe the “size” of a box: given an open

connected subset D of Rd that contains 0 its external radius and internal radius are

defined by

Rext(D) = infR > 0 : BR(0) ⊇ D,(3.1.1)

rint(D) = supr > 0 : Br(0) ⊆ D,(3.1.2)

respectively. Notice that if B = D × C is a box, then Rext(D) < R(C) and rint(D) < r(C).

The recognition radius of a local vertical C in ΩX is defined by

(3.1.3) Rrec(C) = infR > 0 : CY,R ⊆ C for all Y ∈ C,

where CY,R stands for

Z ∈ ΩX | Z ∩BR(0) = Y ∩BR(0).

Notice that if χC is the characteristic function of C, then χC satisfies:

Y ∩BRrec(C)(0) = Z ∩BRrec(C)(0) implies χC(Y ) = χC(Z).

(this means that χC is strongly Y -equivariant in the leaf of Y for every Y ∈ ΩX and Rrec(C)

turns out to be its range (cf. Chapter 5)). Alternatively, this means that to decide whether

a Delone set Y ∈ ΩX belongs to C or not depends only on the Rrec(C)-patch of Y centered

at 0.

3.2 Box decompositions and derived tilings

Let B = B1, . . . , Bt be a finite collection of boxes in ΩX . We say that B is a box decom-

position of ΩX if B is pairwise disjoint and ∪ti=1Bi = ΩX . If for each i1, . . . , t the local

vertical Ci is chosen as the base of Bi and Di × Ci is the canonical parametrization of Bi,

then we define the base of B as C = ∪ti=1Ci and we write B = Di × Citi=1.

40


Let B = Di × Citi=1 be a box decomposition. The following three notions associated to

B will be useful in the sequel:

• the external radius of B is defined by

Rext(B) = maxi∈1,...,t

Rext(Di);

• the internal radius of B is defined by

rint(B) = mini∈1,...,t

rint(Di);

• the recognition radius of B is is defined by

Rrec(B) = maxi∈1,...,t

Rrec(Ci).

In order to describe a finite collection of boxes B, it is useful to study how each box cuts the

leaves of ΩX . As we will see, if the collection B is a box decomposition, then the way each leaf

cuts the boxes defines a decorated tiling (each tile carries a color and a puncture,c.f. Section

2.3) , the so-called derived tiling of the leaf. Let us first describe how a leaf cuts a finite

collection of boxes B = Di×Citi=1, which does not necessarily define a box decomposition.

For each Y ∈ ΩX and i ∈ 1, . . . t define

Di(Y ) = v ∈ Rd | Y − v ∈ Bi.

Since Ci ⊂ Bi, it follows that RCi(Y ) ⊂ Di(Y ). Moreover, it is not difficult to check that

Di(Y ) = Di +RCi(Y ),

and for each i ∈ 1, . . . t and v ∈ RCi(Y ), the set Di+v is a connected component of Di(Y ).

Finally, we define

TB(Y ) = (Di + v, v, i) : i ∈ 1, . . . , t, v ∈ RCi(Y ).

The next proposition follows directly from the definitions, Lemma 2.5.1 and Proposition

2.3.1.

Proposition 3.2.1. Let B be a finite collection of boxes. Then B is a box decomposition if

41


and only if TB(Y ) is a tiling for every Y ∈ ΩX . In such case, the tiling TB(Y ) is aperiodic

and repetitive for every Y ∈ ΩX and is called the derived tiling of B at Y .

The former Proposition suggests that if the Delone set X is the set of punctures of a given

tiling T as described in Section 2.3.1, then there is a box decomposition having T as derived

tiling at X. This is called the canonical box decomposition associated to T . In the case,

that the Delone set X is not associated to a tiling, the canonical box decomposition will

be the box decomposition having the Voronoi tiling of X as derived tiling.

3.3 Voronoi box decompositions and Derived Voronoi

tilings

In this section we give a method to construct box decompositions based on Voronoi tilings.

Several similar constructions appear in the literature (see e.g. [Bellissard 06, Benedetti 03,

Priebe 97, Besbes 08]). Here we follow [Bellissard 06].

Let C be a local vertical and k ≥ Rrec(C) + 2R(C). In the first step we define a finite

partition of C. Since ΩX is minimal, Theorem 2.2.1 implies that

Ak,C = (Y ∩Bk(0)) : Y ∈ C

is finite. Thus, we write Ak,C = p1, . . . ,pt, where t ∈ N. For each i ∈ 1, . . . , t, consider

(3.3.1) Ci = Cpi = Y ∈ ΩX | Y ∩Bk(0) = pi.

From the choice of k it follows that Ci is a subset of C for each i ∈ 1, . . . , t. Thus

C1, . . . , Ct is a partition of C by clopen sets.

In the second step, for each each i ∈ 1, . . . , t we construct a box Bi based at Ci using

Voronoi cells. The idea is to set Di to be the interior of V0(RC(Y )), where Y belongs to Ci.

Thus, we need to show that Di does not depend on Y ∈ Ci. Indeed, if we denote by nY the

set of neighbors of 0 in RC(Y ) (see Section 2.3.2), then the Voronoi cell V0(RC(Y )) satisfies

V0(RC(Y )) =

y ∈ Rd | ‖y‖ ≤ min

r∈nY‖y − r‖

.

42


Hence, Lemma 2.3.2 implies that V0(RC(Y )) is defined by RC(Y ) ∩ B2R(C)(0). Thus, if

v ∈ B2R(C)(0), then the Rrec(C)-patch of Y centered at v does not depend on Y because

it coincides with pi ∩ BRrec(C)(v) (recall that k ≥ 2R(C) + Rrec(C). This implies that

RC(Y )∩B2R(C)(0) does not depend on Y in Ci and therefore Di is well defined and we define

Bi = Di×Ci. Since, the Voronoi cell V0(RC(Y )) of each Y ∈ C contains no return vector to

C, it is clear that Bi is a box.

Finally, we define

B(C, k) = Di × Citi=1.

The following result ensures that B(C, k) is a box decomposition of ΩX .

Proposition 3.3.1. For k ≥ 2R(C)+Rrec(C) the collection B(C, k) is a box decomposition of

ΩX . We call B(C, k) the Voronoi box decomposition with base C and range k. Moreover,

Rrec(B(C, k)) = k, rint(B(C, k)) = r(C) and Rext(B(C, k)) = R(C).

Proof. By Proposition 3.2.1 it suffices us to show that for every Y ∈ C the collection

TB(C,k)(Y ) is a tiling. Indeed, for each i ∈ 1, . . . , t and v ∈ RCi(Y ) we have Di + v =

V0(RC(Y − v)) + v = Vv(RC(Y )). Thus

TB(C,k)(Y ) = (Vv(RC(Y )), v, i) : i ∈ 1, . . . , t, v ∈ RCi(Y ),

which is clearly a decorated version of the Voronoi tiling of RC(Y ). The “moreover” part is

clear from Lemma 2.3.2 and the construction of the Ci’s.

In Figure 3.1 we observe the derived tiling of the Voronoi Box decomposition of the set of

Penrose punctures.

We end this section by comparing this construction with the construction of derived-tilings

appearing in [Priebe 97, Besbes 08]. Given an aperiodic repetitive Delone set X and ΩX its

hull, consider the increasing sequence of patches (pn)n∈N, where pn = X ∩ Bn(0) for all

n ∈ N, and their associated cylinders

Cpn = Y ∈ ΩX | Y ∩Bn(0) = pn.

For each n ∈ N, define kn = 2R(Cpn) + Rrec(Cpn) and construct, by applying Proposition

3.3.1, the Voronoi box decomposition Bn = B(Cpn , kn). The derived tiling T (Bn, X) coincides

with the derived-tiling of pn as defined in [Priebe 97, Besbes 08].

43


Figure 3.1: Derived tiling of the Voronoi Box decomposition with base equal to the cylinderof a white-star in the Penrose tiling

3.4 Tower systems

In this section we use box decompositions to describe in a hierarchical way the “repetitive

structure” on the hull ΩX of an aperiodic repetitive Delone set X. The method we will de-

scribe is a generalization of the construction of Kakutani-Rohlin partitions of minimal Cantor

systems (see e.g. [Durand 99]) and has been first introduced in [Benedetti 03, Bellissard 06].

Let B = Dj×Cjtj=1 and B′ = C ′i×D′it′i=1 be two box decompositions of ΩX and denote

by C and C ′ their respective bases. We say that B′ is zoomed out of B if for every box B′

in B′ the following conditions hold:

Z-1) for each vertical V in B′ there are j ∈ 1, . . . , t and v ∈ Dj such that

V ⊆ Cj − v;

Z-2) the boundary ∂B′ is included in ∪B∈B∂B;

44


Z-3) there exists B ∈ B such that B ∩B′ 6= ∅ and the boundary of B′ does not intersect the

boundary of B ∩B′.

The former definition is a natural generalization of nested Kakutani-Rohlin towers. In par-

ticular, they allow to describe the hierarchy between B′ and B by a matrix M = (mi,j : 1 ≤i ≤ t′, 1 ≤ j ≤ t) whose coefficients are defined by

mi,j = #v ∈ D′i | C ′i − v ⊆ Cj.

Notice that if B′ is zoomed out of B, then, in terms of derived tilings, this means that each

tile of the (non-decorated) tiling TB′(Y ) is a patch of TB(Y ) for every Y ∈ ΩX . From this it

follows that

(3.4.1) vol(D′i) =t∑

j=1

mi,j vol(Dj) for all i ∈ 1, . . . , t.

A tower system is a sequence of box decompositions (Bn)n∈N =(Dn,i × Cn,itni=1

)n∈N

such that:

TS 1) The sequence (Cn)n∈N is decreasing and diam(Cn) → 0 as n → +∞, where Cn =

∪tni=1Cn,i.

TS 2) For every n ∈ N the box decomposition Bn+1 is zoomed out of Bn.

Associated to a tower system there is a sequence of non-negative integer matrices (Mn)n≥1,

where Mn = (m(n)i,j ; 1 ≤ i ≤ tn, 1 ≤ j ≤ tn−1) is a matrix of dimension tn × tn−1. The

coefficients m(n)i,j are defined by

m(n)i,j = #v ∈ Dn,i |Cn,i − v ⊆ Cn−1,j.

The following theorem is proved in [Bellissard 06]. We observe that a proof can be ob-

tained by applying Lemma 3.4.3 below, which is a strengthened version of Theorem 2.441 in

[Bellissard 06]. Since the argument is essentially contained in the proof of Theorem 3.5.1 in

Section 3.5, we omit the proof.

Theorem 3.4.1. Every minimal Delone system ΩX possesses a tower system.

1The “forcing the border” condition appearing in [Bellissard 06] follows easily from the choice of k

45


We will need the following fact:

Fact 3.4.2. Suppose that B′ = Ej ×Cjt′j=1 is a Voronoi box decomposition. Then, for each

j ∈ 1, . . . , t there exists a set Ej ⊂ Rd satisfying Ej ⊆ Ej ⊆ Ej, and the collection

B = Γ(Ej × Cj)t′

j=1

is a partition of ΩX .

We give the definition of Ej and left the proof of Fact 3.4.2 to the reader. Consider the

derived tiling T ′ of B′ at some point Y ∈ ΩX . We define in an arbitrary, but consistent way,

for each t ∈ T ′,

t∗ = p ∈ Rd | p+ (ε, ε2, . . . , εd) ∈ t for all sufficiently small ε > 0.

For each j ∈ 1, . . . , t′ we observe that Ej + v coincides with a tile t of T ′ for some v ∈ Rd

and let Ej = t∗ − v. We leave to the reader to check the properties.

Lemma 3.4.3. Let B be a box decomposition with base C and C ′ be a clopen subset of C

such that r(C ′) ≥ 2Rext(B). Take k′ ∈ N satisfying

k′ ≥ max2R(C ′) +Rrec(C′), R(C ′) +Rext(B) +Rrec(B).

Then, there is a box decomposition B′ = D′j × C ′jt′j=1 with base C ′ that is zoomed out of B

and satisfies Rrec(B′) ≤ k′, rint(B′) ≥ r(C ′)−Rext(B) and Rext(B′) ≤ R(C ′) +Rext(B).

Proof. Let B = Di ×Citi=1 and B(C ′, k′) = Ej ×C ′jt′j=1. The proof consists in modifying

the heights of the boxes in B(C ′, k′) to obtain a box decomposition B′ that is zoomed out of

B. We proceed by steps and work with derived tilings.

1) Let Y ∈ ΩX . We “deform” each tile of TB(C′,k′)(Y ) into a tile that is the support of a

patch of TB(Y ). More precisely, for each j ∈ 1, . . . , t′ and v ∈ RC′j(Y ) let Ej be defined

as in Fact 3.4.2 and define pv(Y ) to be the collection of tiles of TB(Y ) whose punctures are

inside Ej + v, i.e.,

(3.4.2) pv(Y ) = Di + w : i ∈ 1 . . . , t, w ∈ RCi(Y ) ∩ (Ej + v).

Fact 3.4.2 implies that the collection T = pv(Y ) : v ∈ RC′(Y ) is pairwise disjoint and any

tile of TB(Y ) belongs to an element of it. This means that if we identify each pv(Y ) to its

46


support, then T will be a tiling of Rd, provided each pv(Y ) is homeomorphic to a closed ball.

However, it is not necessarily the case.

2) Modify pv(Y ) to make it homeomorphic to a closed ball (We only sketch this step). For

each v ∈ RC′(Y ) let qv(Y ) be the path-connected component of supp pv(Y ) that contains

v. It is not difficult to check that each hole of qv(Y ) is tiled by tiles in TB(Y ), and that

these tiles form the path-connected components of supp pv′(Y ) that do not contain v′ for any

v′ ∈ RC′(Y ). Thus, by identifying each hole with the patch of TB(Y ) that covers it, letting

qv(Y ) = qv(Y ) ∪ t ∈ h : h is a hole of qv(Y )

we obtain a tiling

T (Y ) = (supp qv(Y ), v, j) : j ∈ 1, . . . , t′, v ∈ RC′j(Y ).

of Rd.

3) For all Z ∈ ΩX and w ∈ RC′j(Z) we have qw(Z) = qv(Y ) − v + w. Indeed, from the

construction of B(C ′, k′) it follows that

(Y − v) ∩Bk′(0) = (Z − w) ∩Bk′(0).

Since Ej ⊆ Ej, Lemma 2.3.2 implies that Ej ⊆ BR(C′)(0). Hence supp pv(Y ) is contained in

the closed ball of radius R(C ′) + Rext(B) around v. Hence, by the definition of Rrec(B) we

see that for every i ∈ 1, . . . , t the set

(3.4.3) RCi(Y ) ∩ (Ej + v)

depends only on the (R(C ′) + Rext(B) + Rrec(B))-patch around v. Since k′ ≥ R(C ′) +

Rext(B) +Rrec(B), this implies that RCi(Y )∩ (Ej +w) = (RCi(Y )∩ (Ej + v))− v+w. Hence

pw(Z) = pv(Y )− v + w and the desired equality now follows.

4) Set D′j = supp qv(Y ), where Y ∈ ΩX and v ∈ RC′j(Y ). Since T is a tiling, Proposition 3.2.1

implies that B′ = D′j×C ′jt′j=1 is a box decomposition. It is clear from the construction that

B′ satisfies conditions Z-1) and Z-2). Condition Z-3) follows easily from r(C ′) ≥ 2Rext(B)

and rint(B′) ≥ r(C ′)−Rext(B).

In Figure 3.2 we observe a region of the derived tiling obtained using the construction of

47


the previous Lemma applied to the cylinder defined by the white-star of Figure 3.1.

Figure 3.2: The derived tiling of a box decomposition zoomed out of the canonical Penrosetiling obtained by the ”surgery” process to the Voronoi box decomposition illus-trated in Figure 3.1.

3.5 Tower systems for linearly repetitive Delone sets

In this section we suppose that X is linearly repetitive with constant L > 0, and we use the

estimates produced by Lemma 3.4.3 to obtain a tower system for ΩX satisfying some addi-

tional (and nice) properties. The idea is to consider a decreasing sequence of local verticals

(Cn)n∈N such that ∩n∈NCn = X and apply Lemma 3.4.3 to construct a tower system. To ob-

tain the additional properties we will consider a special sequence such that diam(()Cn)→ 0

exponentially. More precisely, we prove

Theorem 3.5.1. Define K = 6L(L + 1)2 and sn = Kns0 for all n ∈ N and consider the

sequence (Cn)n∈N of local verticals where each Cn is defined by

(3.5.1) Cn = CX,sn = Y ∈ ΩX | Y ∩Bsn(0) = X ∩Bsn(0).

48


Then, there is a tower system (Dn,i ×Cn,itni=1)n∈N of ΩX , where Cn = ∪iCn,i for all n ∈ N,

that satisfies the following properties:

1) for every n ∈ N, Cn+1 ⊆ Cn,1,

2) there exist constants 0 < K1 < 1 < K2 such that for every n ≥ 0,

K1sn ≤ rint(Bn) < Rext(Bn) ≤ K2sn+1.

3) for every n ∈ N, the matrix Mn has strictly positive coefficients,

4) the matrices Mnn∈N are uniformly bounded in size and coefficients, and

5) for each Y ∈ ΩX we have that each point of Y lies in the interior of a tile of Tn(Y ) for

every n ∈ N.

For the proof we will need the following two lemmas.

Lemma 3.5.2. For every n ∈ N we have

(3.5.2)sn

2(L+ 1)≤ r(Cn) < R(Cn) ≤ Lsn.

Proof. The last inequality in (3.5.2) follows from the linear repetitivity of X . The first

inequality in (3.5.2) is a direct consequence of the following fact: if x, y ∈ X satisfy x 6= y

and (X − x) ∩ Bs(0) = (X − y) ∩ Bs(0) for s > 0, then ‖x− y‖ > s/L. We prove this fact

by contradiction. Suppose that there are s > 0 and x, y ∈ X with 0 < ‖x − y‖ ≤ s/L that

satisfy

(3.5.3) (X − x) ∩ Bs(0) = (X − y) ∩ Bs(0).

Take u ∈ X and let pu = X ∩ Bs/L(u). The linearly repetitivity of X implies that there

is u′ ∈ Bs(x) such that pu′ = X ∩ Bs/L(u′) = pu − u + u′. From (3.5.3), it follows that

u′ + (y − x) ∈ X, which implies that u′ + (y − x) ∈ pu′ since ‖x − y‖ ≤ s/L. Hence

u + (y − x) ∈ pu ⊂ X. Since u was chosen arbitrarily, this means that y − x is a period of

X, which is clearly a contradiction since X is aperiodic.

Lemma 3.5.3. Let (Bn)n∈N be a tower system of ΩX . Write Bn = Dn,i × Cn,itni=1 for each

n ∈ N. Then the following assertions hold:

(a) If MX(Rrec(Bn)) ≤ rint(Bn+1) for every n ∈ N, then the coefficients of the transition

matrices Mn are strictly positive.

49


(b) If for every n ∈ N the coefficients of the transition matrix Mn are strictly positive, and

supn∈NRext(Bn+1)/rint(Bn) is finite, then Mnn≥0 is finite set.

Proof. By the definition of Mn, the assertion in (a) is equivalent to say that each slice in

a box in Bn+1 intersects all the boxes in Bn. Thus, given a box (Dn+1,i, Cn+1,i) in Bn+1

it is enough to show that the slice h(Dn+1,i × Y0) around a point Y0 ∈ Cn+1,i intersects

Cn,j for every j ∈ 1, . . . , tn. By the assumption in (a), Dn+1,i contains an occurrence of

every Rrec(Bn)-patch of Y . As the sets Dn,j are determined a Rrec(Bn)-patch this proves

(a). To see (b), we observe that every Dn+1,i contains at least one copy of Dn,j for each

j ∈ 1, . . . , tn. As each Dn,j contains a ball of radius rint(Bn) and Dn+1,i is included in a

ball of radius Rext(Bn+1), we have tn vol(Brint(Bn)(0)) ≤ vol(BRext(Bn+1)(0)) and it follows that

tn ≤ P d, where P = supn∈NRext(Bn+1)/rint(Bn). This gives the bound on the sizes of the

transition matrices. Now observe that every Dn+1,i contains m(n+1)ij copies of Dn,j for each

j. By the same argument as before, we deduce that m(n+1)ij ≤ P d, which gives the bound on

the coefficient of the transition matrices.

Proof of Theorem 3.5.1. For each n ∈ N we define kn = 2R(Cn) + sn. We construct a tower

system by induction on n and use Lemma 3.4.3 as a key part in the induction step. The

estimates given by Lemma 3.4.3 are then used to prove properties 1) to 4).

For the basis step, we set B0 = B(C0, k0). Since s0 ≥ Rrec(C0) by definition of C0,

k0 ≥ 2R(C0)+Rrec(C0). Thus Proposition 3.3.1 ensures that B(C0, k0) is a box decomposition

with Rrec(B0) ≤ k0 and Rext(B0) = R(C0). By permuting the indices if necessary, we obtain

that Y ∈ C0,1.

For the inductive step, we fix n ∈ N and suppose that Bn−1 is a box decomposition that

satisfies:

Rrec(Bn−1) ≤ kn−1(3.5.4)

Rext(Bn−1) ≤ L

K − 1sn.(3.5.5)

We need to show that Bn−1 and kn satisfy the hypotheses of Lemma 3.4.3. That is, we

50


need to show that

kn ≥ 2R(Cn) +Rrec(Cn),(3.5.6)

kn ≥ R(Cn) +Rext(Bn−1) +Rrec(Bn−1),(3.5.7)

r(Cn) ≥ 2Rext(Bn−1).(3.5.8)

(3.5.6) is clear since sn ≥ Rrec(Cn) by the definition of C0. Now we check (3.5.7). First

replace the right-hand side of (3.5.2) (applied for n− 1) into the definition of kn−1 and then

replace sn = Ksn−1 in the result to obtain

(3.5.9) kn−1 ≤2L+ 1

Ksn.

Hence replacing (3.5.9) into (3.5.4) and then adding (3.5.5) we obtain

(3.5.10) Rext(Bn−1) +Rrec(Bn−1) ≤(

L

K − 1+

2L+ 1

K

)sn.

An easy computation shows that the right side of (3.5.10) is smaller or equal than sn and

thus (3.5.7) follows from the definition of kn. Finally, (3.5.8) follows from replacing the

left-hand side of (3.5.2) into (3.5.5).

Hence Lemma 3.4.3 implies there is a box decomposition Bn = (Dn,i, Cn,i)tni=1 zoomed

out of Bn−1 that satisfies

Rrec(Bn) ≤ kn(3.5.11)

r(Cn)−Rext(Bn−1) ≤ rint(Bn)(3.5.12)

Rext(Bn) ≤ R(Cn) +Rext(Bn−1).(3.5.13)

To finish the inductive step, it remains to show that

(3.5.14) Rext(Bn) ≤ L

K − 1sn+1.

This is proved easily by replacing (3.5.2) and (3.5.5) into (3.5.13). Hence, applying the

induction above we obtain a sequence of box decompositions (Bn)n∈N such that Bn is zoomed

out of Bn−1 and satisfies (3.5.11),(3.5.12) and (3.5.13).

Finally we check properties 1) to 4). After permuting indices we have that Y ∈ Cn,1 for

every n ∈ N. Hence, since sn+1 > kn ≥ Rrec(Bn) it follows that Cn+1 = CY,sn+1 ⊆ Cn,1 and

51


1) holds. Replacing (3.5.2) and (3.5.5) into (3.5.12) we obtain

(3.5.15) rint(Bn) ≥ sn

(1

2(L+ 1)− L

K − 1

)for all n ∈ N.

Hence, (3.5.15) and (3.5.14) imply that property 2) holds with K1 = 12(L+1)

− LK−1

and

K2 = LK−1

. We now check that property 3) holds. By Lemma 3.5.3 (a) it suffices us to show

that

(3.5.16) Lkn ≤ rint(Bn+1) for all n ∈ N.

Indeed, combining (3.5.9) into (3.5.15) yields

K1K

2L+ 1kn ≤ rint(Bn+1).

By the definition of K, we have that K1K/(2L+1) ≥ L and (3.5.16) holds. Finally, we check

that (4) holds. Indeed, dividing (3.5.5) by (3.5.15) and then replacing sn+2 = K2sn into the

result yields

(3.5.17)R(Bn+1)

r(Bn)≤ K2K−1

1 K2

for every n ∈ N, which means that R(Bn+1)/r(Bn) is uniformly bounded in n ∈ N. Hence,

applying Proposition 3.5.3 (b) yields the desired conclusion.

52

4 On the rate of convergence of patch

frequencies for linearly repetitive

Delone sets

4.1 Motivations and results

Let M be a compact metric space, T : M → M a homeomorphism, and µ an ergodic T -

invariant probability measure. Birkhoff’s Ergodic Theorem states that for every Borel set A

the quantity

(4.1.1)1

Ncard

k ∈ 0, . . . , N − 1 : T k(x) ∈ A

→ µ(A) a.e. as N → +∞.

It is natural to study the rate of convergence in (4.1.1). However, in general no such rate

exists (see e.g. [Halasz 76, Petersen 89] for details). An alternative formulation is to restrict

the study to a subclass of Borel sets. For instance, in [Halasz 76] it is shown that if µ(A) is

a measurable eigenvalue of the system (M,µ, T ), then

(4.1.2) | 1N

cardk ∈ 0, . . . , N − 1 : T k(x) ∈ A

− µ(A)| = O(N−1) a.e.,

and this is the best convergence rate possible in this setting.

In this chapter we will suppose that X is an aperiodic linearly repetitive Delone set. For

every r-patch p, r > 0, every Y ∈ ΩX and a region D ⊆ Rd, we define

np(D, Y ) = cardy ∈ Y ∩D | Y ∩Br(y) is a translated of p,

i.e., np(D, Y ) is the number of patches of Y whose centers belong to D and are translates of

53

4 On the rate of convergence of patch frequencies for linearly repetitive Delone sets

p. Since X has uniform pattern frequencies (see Section 2.6 for details), the following limit

(4.1.3) limN→+∞

|np(BN , Y )|vol(BN)

= freq(p)

exists for every p and Y ∈ ΩX , which gives a higher-dimensional version of (4.1.1). Moreover,

the following result of Lagarias and Pleasants (see [Lagarias 03]) gives a bound in the rate

of convergence of (4.1.3):

Lagarias-Pleasant’s Theorem. Let X be a linearly repetitive Delone set and p ⊂ X be a

patch of X. Then, there exists 0 < δ < 1 such that∣∣∣∣np(BN , X)

vol(BN)− freq(p)

∣∣∣∣ = O(N−δ),

where BN = [−N,N ]d.

The objective of this chapter is to improve the theorem of Lagarias and Pleasants. The

discussion at the beginning of this section suggests that, at least in the case d = 1, the fastest

convergence is by obtained by taking δ = 1. Thus, we say that np converges fast to freq(p)

if

(4.1.4) |np(BN , X)− vol(BN) freq(p)| = O(Nd−1).

Through this chapter (Dn,i × Cn,itni=1)n∈N will be the tower system of ΩX obtained by

applying Theorem 3.5.1 to X. Recall that this system satisfies the following properties:

(1) The sequence of bases (Cn)n≥0 is a decreasing sequence in ΩX .

(2) The transition matrices Mn are strictly positive and uniformly bounded in size and

coefficients.

(3) There exist a sequence of strictly positive numbers (sn)n≥0 and positive constants K,K1

and K2 such that for every n ∈ N

sn+1 = Ksn,

and

B(0, K1sn) ⊆ Dn,i ⊆ B(0, K2sn+1).

(4) For all n ∈ N and Y ∈ ΩX no point of Y lies on the boundary of a tile of Tn(Y ).

54


For each r > 0 we define n0(r) as the smallest integer n such that for every i ∈ 1, . . . , tnand v ∈ Dn,i we have that

Y, Z ∈ Cn,i − v implies Y ∩Br(0) = Z ∩Br(0).

It is easy to check that for every r-patch p and n ≥ n0(r) the quantity np(Dn,i, Y ) does not

depend on Y ∈ Cn,i for every 1 ≤ i ≤ tn. Thus, we write np(Dn,i) instead of np(Dn,i, Y ).

Finally, we are able to state the main result of this chapter:

Theorem 4.1.1. Let X be a linearly repetitive aperiodic Delone set in Rd. If for an r-patch

p we have

(4.1.5)∞∑

n=n0(r)

s1−dn+1 max

i|np(Dn,i)− vol(Dn,i) freq(p)| <∞,

then np(BN , Y ) converges fast to freq(p) as N goes to infinity for every Y ∈ ΩX .

In the next section we will give a precise bound for the error term in (4.1.4) in terms of

the tower system (c.f. Lemma 4.2.1) from which Theorem 4.1.1 follows directly. Then the

bound is used in Section 4.3 to give a proof of Lagarias and Pleasants theorem.

4.2 Estimating the error term

Let Y be in ΩX . Recall that the derived tiling of Dn,i × Cn,itni=1 and Y is defined by

(4.2.1) Tn(Y ) = (Dn,i + v, v, i) : i ∈ 1, . . . , tn, v ∈ RCi(Y ).

Given N ∈ N, let n1(N) be the largest integer n such the set t ∈ Tn(Y ) | t ⊆ BN is not

empty.

Lemma 4.2.1. There exists M > 0 such that for every Y ∈ ΩX and r > 0 we have

|np(BN , Y )−Nd freq(p)| ≤ MNd−1

n1(N)∑n=n0(r)

s1−dn+1 max

i

∣∣np(Dn,i)− vol(Dn,i) freq(p)∣∣+ 1

for every r-patch p of Y and every N ∈ N.

55


For the proof we need the following easy estimation:

Lemma 4.2.2. There exists M ≥ 0 such that for every Y ∈ ΩX and every n ∈ N, n ≤n1(N) + 1, the number of tiles of Tn(Y ) that intersect the boundary of BN is bounded above

by MNd−1s1−dn .

Proof. Fix Y ∈ ΩX , N ∈ N and n ∈ N, n ≤ n1(N) + 1. Denote by A the set of tiles of Tn(Y )

that intersect ∂BN . By property (3), each tile in A contains a ball of radius K1sn and is

included in a ball of radius K2sn+1. Since tiles in A do not overlap, this implies that

(4.2.2) Kd1s

dn vol(B1(0))#A ≤ vol((∂BN)+2K2sn+1),

where (∂BN)+2K2sn+1 = x ∈ Rd | dist(x, ∂Bn) ≤ 2K2sn+1. It is easy to check that

(4.2.3) vol((∂BN)+2K2sn+1) ≤ 2d+2dK2sn+1(2K2sn+1 +N)d−1.

By definition of n1(N), there is a tile in Tn1(Y ) that is included in BN . By property (3), this

tile contains a ball of radius K1sn1(N) and hence

(4.2.4) K1sn1(N) ≤ N.

It follows that K1sn+1/K2 ≤ N since (sn)n∈N is increasing. Hence (4.2.3) implies

(4.2.5) vol((∂BN)+2K2sn+1) ≤ 2d+2K2

(2K2K2

K1

+ 1

)d−1

Nd−1sn+1.

The conclusion now follows from (4.2.2) and (4.2.5) with M being defined by

M = 2d+2dK2(Kd1 vol(B1(0)))−1K

(2K2K2

K1

+ 1

)d−1

.

Proof of Lemma 4.2.1. We fix Y ∈ ΩX , N ∈ N, r > 0 and denote n0 = n0(r) and n1 = n1(N).

The idea of the proof is to decompose BN into smaller pieces that are tiles of Tn(Y ) for some

n ∈ n0, . . . , n1. Since the tiles of Tn(Y ) are tiled by tiles of Tm(Y ) for all m ≤ n, we ask

this decomposition to contain tiles as big as possible.

56


Figure 4.1: Description of the patches of different levels using the box decomposition of Figure3.2.

More precisely, we define

Pn1 = t ∈ Tn1(Y ) | t ⊆ BN and

Qn1 = suppPn1 .

For n ∈ n0, . . . , n1 − 1, Qn and Pn are defined recursively as follows

Pn =t ∈ Tn(Y ) | t ⊆ BN \Qn+1

,

Qn = suppPn

It is easy to check that for every n ∈ n0, . . . , n1 each tile of Pn lies inside a tile of Tn+1(Y )

that intersects the boundary of BN . See Figure 4.1. It follows from Lemma 4.2.2, Property

(2) and Property (4) that

(4.2.6) #Pn ≤MαdNd−1s1−dn+1 for all n ∈ n0, . . . , n1,

where α > 0 is a uniform bound for the coefficients and the size of the transition matrices.

57


Let W = BN \ ∪n1n=n0

Qn. Since the Qn’s do not overlap, we have

vol(BN) =

n1∑n=n0

∑t∈Pn

vol(t) + vol(W ).(4.2.7)

From property (4) it follows that

np(BN , Y ) =

n1∑n=n0

∑t∈Pn

np(t, Y ) + np(W,Y ).(4.2.8)

By definition of Tn(Y ), every t ∈ Pn can be written as Dn,i + v, where i ∈ 1, . . . , tn and

v ∈ RCi(Y ), and since n ≥ n0, then np(t, Y ) = np(Dn,i). Hence

(4.2.9) |np(t, Y )− vol(t) freq(p)| ≤ maxi∈1,...,tn

|np(Dn,i, Y )− vol(Dn,i) freq(p)|

for every t ∈ Pn. Thus, from (4.2.8), (4.2.7) and (4.2.9) we obtain

|np(BN , Y )− freq(p) vol(BN)| ≤n1∑

n=n0

#Pn maxi∈1,...,tn

|np(Dn,i, Y )− vol(Dn,i) freq(p)|

+ |np(W,Y )− vol(W ) freq(p)|.

(4.2.10)

If we suppose that

(4.2.11) |np(W,Y )− vol(W ) freq(p)| = O(Nd−1),

then replacing (4.2.11) and (4.2.6) into (4.2.10) gives the conclusion of the Lemma. Thus, it

remains to prove (4.2.11). Let A0 be the set containing all the tiles of Tn0(Y ) that intersect

the boundary of BN . By definition, W is included in the support of A0. By Lemma 4.2.2,

there exists M > 0 such that #A0 ≤MNd−1. Hence, we get

vol(W ) ≤MNd−1s1−dn0

maxi∈1,...,tn0

vol(Dn0,i)

and

np(W,Y ) ≤MNd−1s1−dn0

maxi∈1,...,tn0

np(Dn0,i),

from which (4.2.11) easily follows.

58


4.3 A proof of Lagarias and Pleasants Theorem

For n,m ∈ N, n > m, define the matrix P (n,m) = MnMn−1 . . .Mm+1 and P (n) = P (n, 0),

where (Mn)n∈N is the sequence of transition matrices (see Section 3.4) associated to the tower

system of ΩX . Observe that the coefficient p(n,m)i,j counts the number of tiles equivalent to

Dm,j that appear in a tile that is equivalent to Dn,i. We will need an exponential mixing

property. This is a standard result (see e.g. [Bressaud 05] or [Coronel 09] where a detailed

proof is given).

Lemma 4.3.1. There exists c ∈ (0, 1) such that for all n ∈ N∗ and m ∈ N

max1≤j≤tm

1≤i≤tm+n

∣∣∣∣∣ p(n+m,n−1)ij

vol(Dn+m,i)− ν(Cn−1,j)

∣∣∣∣∣ ≤ Lν(Ω0X)cm+1.

Denote by ν the transverse invariant measure associated with the unique invariant proba-

bility measure on ΩX . Recall from Chapter 2 that freq(p) = ν(Cp), where

Cp = Y ∈ Ω0X | Y ∩Br(0) = p− x

and x is the center of p.

Lemma 4.3.2. For every r-patch p of X, n ≥ n0 and i ∈ 1, . . . , tn we have

(4.3.1) np(Dn,i)− vol(Dn,i) freq(p) =

tn0∑k=1

np(Dn0,k)(p

(n,n0)ik − ν(Cn0,k) vol(Dn,i)

).

Proof. Property (4) of the tower system (Bn)n∈N implies that

(4.3.2) np(Dn,i) =

tn0∑k=1

np(Dn0,k)p(n,n0)ik .

From the definition of n0 it follows that Cp is partitioned by local verticals that are translates

of the bases of the boxes in Bn0 . Then it is easy to check that

(4.3.3) ν(Cp) =

tn0∑k=1

np(Dn0,k)ν(Cn0,k).

Hence the conclusion follows after an easy computation involving (4.3.3) and (4.3.2).

59


Using Lemma 4.3.2 and Lemma 4.1.1 we obtain the following condition to ensure the

fast-convergence of all patches.

Corollary 4.3.3 ([Lagarias 03, Theorem 6.1]). Let X be a linearly repetitive Delone set in

Rd and r > 0. Then, there exists 0 < δ < 1 such that for every r-patch p of X,

|np(BN + x,X)−Nd freq(p)| = O(Nd−δ)

uniformly in x ∈ Rd.

Proof. Notice that np(BN + x,X) = np(BN , X − x) for every x ∈ Rd. By Lemma 4.3.2 and

4.3.1 we have that

(4.3.4) |np(Dn,i)− vol(Dn,i) freq(p)| ≤ Ctn0C′cn−n0 vol(Dn,i),

By property (3), vol(Dn,i) = O(sdn+1) and by property (4) tn0 is bounded. Thus, from (4.3.4)

we get

(4.3.5) maxi∈1,...,tn

∣∣np(Dn,i)− vol(Dn,i)ν(Cp)∣∣ = O

(cn−n0sdn+1

).

Hencen1∑

n=n0

s1−dn+1 max

1≤i≤tn∣∣np(Dn,i)− vol(Dn,i)ν(Cp)

∣∣ = O

(n1∑

n=n0

sn+1cn−n0

).

In virtue of Lemma 4.2.1, it suffices to show that there exists 0 ≤ γ < 1 such that

n1∑n=n0

sn+1cn−n0 = O(Nγ).

Let β = Kc. By property (3) we get

(4.3.6)

n1∑n=n0

sn+1cn−n0 = sn0+1

n1−n0∑n=0

βn =

(1− βn1−n0+1)/(1− β) if β 6= 1,

n1 − n0 + 1 otherwise.

To conclude we distinguish three cases:

1. β < 1, in which case the right-hand term of (4.3.6) is bounded and the conclusion

clearly follows.

60


2. β = 1: Taking logarithm to (4.2.4) (see the proof of Lemma 4.3.2) we get

logK1s0 + n1(N) logK ≤ logN

and it follows that n1(N) = O(logN).

3. β > 1: Suppose that log(Kc)/ log(K) < γ < 1. From (4.2.4) we get n1(N) ≤ (logN −log s0K1)/ logK. Hence

log

(βn1−n0+1 − 1

β − 1

)≤ (n1 + 1) log β − log(β − 1)

≤(

logN

s0K1

+ logK

)logKc

logK− log(β − 1)

≤ log

(Kγ

rγ0Kγ1 (β − 1)

Nγ

),

which concludes the proof.

61

5 Pattern-equivariance and translation

numbers for maps in the real-line

In this chapter, we study Delone sets of the real line, i.e., we suppose that d = 1. In this

context, a Delone set X will be seen as an increasing sequence xnn∈Z satisfying 2r ≤(xn − xn−1) ≤ 2R with r, R > 0. It is easy to check that the Delone set xnn∈Z has finite

type if and only if the set xn+1− xn : n ∈ Z is finite. In the remainder of this dissertation,

we will use the term quasicrsytal to refer to a repetitive Delone set with uniform patch

frequencies. Unless explicitly mentioned, a quasicrystal will also be supposed aperiodic.

5.1 Introduction: short range potentials and

pattern-equivariant functions

We are interested in the study of the limiting behavior of average displacements for a given

non-decreasing self-map on the real line. More precisely, let f : R→ R be a non-decreasing

continuous map. We study the existence of the following limit

ρ(f, t) = limn→+∞

fn(t)− tn

.

Much is known when the displacement of f , defined by t 7→ f(t) − t is a periodic function,

which is the subject of Poincare’s theory of circle homeomorphisms. In this case, the dis-

placement of f factors through a circle map and it is known that the former limit exists for

every t ∈ R. Moreover, it is independent of t and the convergence is uniform. We refer to

this number as the translation number of f at t.

In this chapter, we focus on the case where the displacement of f is a quasi-periodic

function in the sense we now define. Let X be a repetitive Delone set. A function φ : R→ R

62

5 Pattern-equivariance and translation numbers for maps in the real-line

is called strongly X-equivariant or short-range potential if there exists S > 0, called

the range of φ, such that

(X − t) ∩ [−S, S] = (X − s) ∩ [−S, S]

implies

φ(t) = φ(s).

A continuous function φ : R → R is called X-equivariant if it is the uniform limit of a

sequence of strongly X-equivariant continuous functions. We denote by CX(R) the set of all

X-equivariant functions.

In the case where X is a perfect crystal, we check that short-range functions coincide with

periodic functions. It is a standard fact that a continuous periodic function factors through

a continuous map on the circle. When X is aperiodic and repetitive, then an analogous

correspondence is stated in the following Lemma proved in [Kellendonk 03]:

Lemma 5.1.1. The map C : C(ΩX)→ CX(R) defined by

(5.1.1) C(Φ)(t) = Φ(X − t).

is a bijection. That is to say, every X-equivariant function factors through a continuous

function over ΩX . Moreover, every continuous function in C(ΩX) can be obtained in this

way.

When the displacement of f is X-equivariant we say that f has X-equivariant displacement.

We denote by F+X(R) the set of non-decreasing maps of the real line with X-equivariant

displacement. Lemma 5.1.1 implies that maps in F+X(R) factor through self-maps of ΩX of

the form

Q 7→ Q− Φ(Q), for all Q ∈ ΩX

where Φ is the continuous function factorizing the displacement of f . It follows that these

maps are homotopic to the identity. It is natural to ask whether all maps that are homotopic

to the identity and preserve the orientation can be obtained in this way. This question is

addressed in the following section, see Theorem 5.3.3 in Section 5.3.

Once the answer to this question is given, in Section 5.4, we recall the basic definitions

of translation (rotation) theory adapted to our case and give some results describing the

63


Figure 5.1: Construction of the Fibonacci chain

translation sets for the maps studied.

We end this introduction by giving an example of how to construct short-range potentials.

Example 5.1. The standard examples of quasicrystals on the real line are given by primitive

substitution sequences (for more details see [Durand 99]) like the Fibonacci sequence. This

sequence is constructed by iterating the substitution:a → ab

b → a

starting by the sequence a.a one obtains a bi-infinite sequence wnn∈Z that is a periodic

point (of period 2) of the previous substitution. Given L, S > 0, a Fibonacci quasicrystal

is a Delone set FibL,S = xnn∈Z such that xn+1 − xn is equal to L if wn = a and to S if

wn = b (see figure 5.1). If Ω denotes the continuous hull of Xfib, then it is easy to check

that the canonical transversal of Ω is conjugate with the Fibonacci shift. There are two

important choices for L and S. First, if L = S = 1, then Ω is conjugate with the standard

suspension of the Fibonacci shift. Second, if L = τ := (√

5 + 1)/2 and S = 1, then Xfib is

self-similar (see [Radin 01] for details): there is a homeomorphism ω : Ω → Ω, the so-called

inflation-substitution homeomorphism, which satisfies ω(Xfib−t) = Xfib−τt for every t ∈ R.

A simple way to obtain Fib-equivariant functions consists in choosing two real valued

smooth functions, vL,L, and vS,L with compact support on the interval (−I, I) where 0 <

2I < L and 0 < 2I < S. A strongly Fib-equivariant function φFib can be defined as follows

64


Figure 5.2: A Fib-equivariant function

(see figure 5.2): If θ ∈ (xn − I, xn + I) for some n ∈ Z then we set

φFib(θ) =

vL,L(θ − xn), if |xn − xn−1| = |xn − xn+1| = L,

vS,L(θ − xn), if |xn − xn−1| 6= |xn − xn+1|.

If not, we set φFib(θ) = 0.

5.2 Return times and Poincare maps for local verticals in

Delone spaces

Let ΩX be a minimal Delone system. Recall that a local vertical in ΩX is a clopen subset of

a cylinder VY,S with Y ∈ ΩX and S > 0. We consider local verticals as being cross sections

and hence study their induced maps. Namely, for a local vertical V in ΩX and Y ∈ ΩX , we

define the first entry time of Y to V by

tV (X) = inft > 0 : X − t ∈ V .

When Y ∈ V , we call tV (Y ) the first return time of Y to V . The first return map to V is

the map σV : V → V defined by

σV (Y ) = Y − tV (Y ), Y ∈ V.

The following lemma follows directly from Lemma 2.5.1 (see [Herman 92, Putnam 89] for

similar results in the context of minimal Cantor systems) and gives the basic properties of

such objects:

65


S L SL L S L

S L L S L S L SL L S L

0

S SL LL

tV (Fib− s)s = tV (Fib)

Fib

σV (Fib)

Figure 5.3: First return time and first return maps for Fibonacci quasicrystals. Here V =Y ∈ ΩFib : Y ∩ [−L,L] = −L,L.

Lemma 5.2.1. The function tV : ΩX → R is continuous on ΩX and, when restricted to V ,

it takes only finitely many distinct values. Moreover, the set

RV (Y ) = t ∈ R : Y − t ∈ V

is a Delone set of finite type for every Y ∈ V . Moreover, the map σV is a homeomorphism

and the system (V, σV ) is minimal.

We recall the construction of transverse invariant measures (c.f. Section 2.8.2) adapted to

d = 1. Each Γ-invariant measure µ induces a finite measure νV (= νV (µ)) on each clopen

subset of Ω0X as follows: by Lemma 5.2.1 applied to V , there is a finite number of return

times to V , say tV (V ) = h1, . . . ,hl and define Ei := X ∈ V | tV (X) = hi for each

i ∈ 1, . . . , l. Given a clopen set C in V , the measure νV satisfies

νV (C) =l∑

i=1

1

hiµ(X − t : X ∈ C ∩ Ei, 0 ≤ t < hi).

We check that

(5.2.1)C∑i=i

νV (Ei)hi = 1.

If the system (ΩX ,Γ) is uniquely ergodic, then ν will denote the σΩ0X

-invariant measure

on Ω0X induced by unique Γ-invariant probability measure µ following the procedure we

just described (i.e., ν := νΩ0X

(µ)). It is not difficult to check that in this case, for each

clopen subset V of Ω0X , the system (V, σV ) is uniquely ergodic, and moreover, the unique

σV -invariant probability measure on V is given by ν/ν(V ). The following consequence of

Birkhoff’s Ergodic Theorem will be used in the sequel:

Lemma 5.2.2. Suppose that ΩX is uniquely ergodic. For a clopen set V in Ω0X and Y ∈ V ,

66


define

`V (t, Y ) = cards ∈ [0, t) : Y − s ∈ V .

Then for every clopen set V in Ω0X and Y ∈ V we have

limt→+∞

`V (t, Y )

t= ν(V ).

5.3 Maps on Delone systems that are homotopic to the

identity

Let X be an aperiodic repetitive Delone set and ΩX its continuous hull. The usual order on

R induces an order on ΩX as follows: given Y, Z ∈ ΩX , we say that Y Z if there exists

t ≥ 0 such that Y − t = Z (analogous definitions are given for ≺,,). To save notations,

when Y Z we will write [Y, Z] instead of W ∈ ΩX | Y W Z (the notations [Y, Z),

(Y, Z] and (Y, Z) are analogous).

Let F be a continuous self-map on ΩX . Since ΩX is locally the product of a Cantor set and

an interval, it follows that connected components of ΩX coincide with Γ-orbits (or leaves)

of ΩX . Therefore, F sends each leaf to another (maybe different) leaf. We say that F is

orientation-preserving on ΩX if and only if

Y Z implies F (Y ) F (Z).

We denote by F0(ΩX) the set of all continuous self-maps of ΩX that are homotopic to the

identity. Let F be a map in F0(ΩX) and Ftt∈[0,1] be an homotopy with F0 equal to the

identity on ΩX and F1 = F . For each Y ∈ ΩX , the set Ft(Y ) : t ∈ [0, 1] is connected and

therefore it is contained in the leaf of Y . Aperiodicity implies that there is a unique function

Φ : ΩX → R such that F (Y ) = Y − Φ(Y ) for every Y ∈ ΩX . The function Φ is called the

displacement of F .

Proposition 5.3.1. Let F : ΩX → ΩX be a continuous map and Φ its displacement. For

Y ∈ ΩX , define fY : R→ R by

(5.3.1) F (Y − t) = Y − fY (t), t ∈ R.

If Φ is continuous then the function fY has Y -equivariant displacement.

67


Proof. The map fY is well defined by the aperiodicity of ΩX . Lemma 5.1.1 implies that the

function φ : R→ R defined by φ(t) = Φ(X− t) is continuous and X-equivariant. Notice that

F (X − t) = X − t− φ(t). Thus, by aperiodicity of X one has that φ is the displacement of

fY , and the proof is done.

Conversely, if f : R → R has X-equivariant displacement, then there exists Φ ∈ C(ΩX)

such that the map F : ΩX → ΩX defined by

(5.3.2) Y 7→ Y − Φ(Y ), Y ∈ ΩX

satisfies F (X − t) = X − f(t) for every t ∈ R.

Moreover, we check that F preserves the orientation if and only if f is non-decreasing: this

follows from the density of leaves in ΩX and the continuity of Φ.

The following well-known theorem will be needed.

Theorem 5.3.2. Let M be a metric space and f : M → R a function. If f(M) is bounded,

then f is continuous if and only if

gr(f) = (x, f(x)) : x ∈M

is a closed subset of M × R.

Theorem 5.3.3. Let F : ΩX → ΩX be an orientation-preserving continuous map of the form

Y 7→ Y − Φ(Y ), Y ∈ ΩX

where Φ : ΩX → R. Then, the function Φ is continuous and, in particular, F is homotopic

to the identity on ΩX .

Corollary 5.3.4. There is a one-to-one correspondence between maps in F+0 (ΩX) and maps

in F+X(R).

Proof. let F be a map preserving each leaf and its orientation. By Theorem 5.3.3, F can

be written as F (Y ) = Y − Φ(Y ) where Φ is continuous. It thus suffices to take (τ, Y ) 7→Y − τΦ(Y ) to define the desired homotopy.

68


Proof of Theorem 5.3.3. It is easy to see that

gr(Φ) = (Y, t) ∈ ΩX × R | Y − t = F (Y ),

and since F is continuous, it follows that gr(Φ) is closed. Hence, by Theorem 5.3.2 it suffices

to show that Φ is bounded. Indeed, the sets Bn = Y ∈ ΩX : |Φ(Y )| ≤ n are closed for all

n ∈ N by the continuity of F . Since ∪n∈NBn = ΩX , Baire’s Category Theorem implies there

exists n ∈ N such that Bn has non-empty interior, which means that there is a non-empty

open set U ⊂ ΩX and C > 0 such that |Φ(Y )| < C for every Y ∈ U . Fix Y ∈ U and choose

S > 0 to be large enough such that V = VY,S ⊂ U . By lemma 5.2.1, RV (Y ) is a Delone set

of finite type, so we write RV (Y ) as an increasing sequence (tn)n∈Z. As F preserves the leaf

of Y and its orientation, one has

s + Φ(Y − s) ≤ t + Φ(Y − t)

for every t, s ∈ R such that t ≤ s. It follows that, for every s ∈ R,

(5.3.3) |Φ(Y − s)| ≤ C + supn

(tn+1 − tn) =: C.

Recall that the finite local complexity of RV (Y ) implies that C is finite. Let Z ∈ ΩX . By

minimality of (Ω,Γ), we can choose a sequence (zn)n∈N ⊆ R for which Y − zn converges

to Z. From (5.3.3) it follows that (Φ(Y − zn))n∈N is bounded and thus, by dropping to

a subsequence if necessary, we assume that Φ(Y − zn) converges to φz as n → +∞ with

|φz| ≤ C. It follows that F (Y − zn) = Y − zn−Φ(Y − zn) converges to Z −φz. On the other

hand, by the continuity of F we have that F (Y − zn) converges to Z − Φ(Z) as n → +∞and then the aperiodicity of Z yields that φz = Φ(Z), which implies that Φ is bounded.

5.4 Translation sets for self-maps homotopic to the identity

Let X be a quasicrystal and ΩX its continuous hull. Consider a map F in F+0 (ΩX) with

displacement Φ. For each n ∈ N, we define Φ(n) =∑n−1

i=0 Φ F i .

The translation number of F at Y ∈ ΩX is defined by

(5.4.1) ρ(F, Y ) := limn→+∞

Φ(n)(Y )

n,

69


provided the limit exists. The set of all translation numbers at all points where it exists is

called the translation set of F and denoted by ρp(F ). If for every Y ∈ ΩX , the translation

number of F at Y exists and is independent of Y , then we say that the translation number

of F exists or that F has a translation number denoted by ρ(F ).

We remark that if fX is the map induced by F in the leaf of X (see Proposition 5.3.1)

then for every n ∈ N and every t ∈ R,

fnX(t)− t = Φ(n)(X − t).

This means that in order to study translation sets for maps with pattern-equivariant dis-

placements it suffices to study translation sets for maps in F+0 (ΩX). We also remark that

for a F -invariant ergodic probability measure µF , Birkhoff’s Ergodic Theorem states that

the translation number of F exists at µF -almost every point and is equal to∫

ΩXΦdµF . If

F is uniquely ergodic, then the translation number exists at every point (n fact the limit

in (5.4.1) is uniform) and is equal to∫

ΩXΦdµF , i.e., F has a unique translation number.

The following result states that whenever there is a unique translation number, then the

displacement averages converge for every point and the convergence in (5.4.1) is uniform.

Proposition 5.4.1 (Prop.4.2. in [Geller 99]). The following assertions are equivalent for a

compact F -invariant set K ⊂ ΩX :

(a) The translation set of F |K is ρ.

(b) For every Y ∈ K the translation number of F at Y exists and is equal to ρ.

(c) There is a dense set D ⊆ K such that for every Y ∈ K

limn→+∞

supY ∈D

∣∣∣∣ 1nΦ(n)(Y )− ρ∣∣∣∣ = 0.

(d) (c) is satisfied with D = K.

Now we are going to describe the translation set of F . It is plain to check that if F has

fixed points then 0 belongs to the translation set of F . So we will assume for the moment

that F has fixed points. Since fixed points of F coincide with the zeros of Φ, it follows that

Φ has zeros. Thus, we have two cases: either Φ changes signs or Φ does not change signs.

In the first case, the following result gives a complete description of the translation set of F

and the orbit of every point in ΩX . Its proof is based on the Intermediate Value theorem.

70


Proposition 5.4.2. If Φ changes signs then the translation set of F is the singleton 0 and

every point converges under iteration by F to a fixed point.

Proof. Let Y and Z ∈ ΩX be such that Φ(Y ) < 0 < Φ(Z). Since Φ is continuous, we

can find neighborhoods U of Y and V of Z such that Φ is negative in every point of U

and positive in every point of V . By the minimality of (Ω,Γ), there is K > 0 such that

∪t∈[−K,0]ΓtU = ∪t∈[0,K]ΓtU = ∪t∈[−K,0]ΓtV = ΩX ∪t∈[0,K] ΓtV = ΩX . It follows that for

every W ∈ ΩX there exist −K < t1 < t2 < 0 < t3 < t4 < K such that Φ(W − ·) changes

signs between t1 and t2 and also between t3 and t4. Hence, applying the Intermediate Value

Theorem one deduces that there exist −K < s′ < 0 < t′ < K such that W − t′ and W − s′

are fixed points of F . Let fW be the map defined by Proposition 5.3.1. We check that t′ and

s′ are fixed points of fW and since fW is non-decreasing, the interval [s′, t′] is fW -invariant.

It follows that the sequence (fnW (0))n∈N is monotone and bounded. Since Φ(n)(W ) = fnW (0)

for every n ∈ N, it follows that ρ(F,W ) = 0. Moreover, the sequence (fnW (0))n∈N converges

to t0 ∈ [s′, t′] , which is necessarily a fixed point of fW . By (5.3.1) and the continuity of the

translation action, we see that W − t0 is a fixed-point of F and that F n(W ) converges to

W − t0.

Thus, in the remainder of this section Φ does not change signs. To fix ideas, we will assume

that inf Φ ≥ 0. The case sup Φ ≤ 0 is left to the reader and may be treated in an analogous

way.

We define

Ω+X = Y ∈ ΩX : lim sup

kΦ(k)(Y ) = +∞

and set ΩfpX = ΩX \ Ω+

X . We easily check that

• Ω+X is F -invariant,

• the translation set of F restricted to ΩfX is 0 when Ωfp

X is not empty,

• ΩfpX contains the fixed points of F . Moreover Ωfp

X is not empty if and only if F has fixed

points.

Thus, we restrict ourselves to study the translation set of F restricted to Ω+X .

Theorem 5.4.3. Let F be a map in F+0 (ΩX) and Φ be its displacement. If inf Φ ≥ 0 then

there exists ρ ≥ 0 such that for µ-almost every Y ∈ Ω+X , the translation number ρ(F, Y ) exists

and is equal to ρ.

71


Proof of Theorem 5.4.3. Let (Cn)n∈N be a decreasing sequence of local verticals in Ω0X with

diam(Cn)→ 0 as n→ +∞. For the moment, we fix n ∈ N, set σn to denote the first return

map to Cn and define kn : Cn → R ∪ +∞ by

(5.4.2) kn(Y ) = maxk ≥ 0 : F k(Y ) σn(Y ), Y ∈ Cn.

Let Y ∈ Ω+X and define the sequence (Y`)`∈N recursively by:

Y0 =

Y − tCn(Y ) if Y 6∈ Cn,

Y if Y ∈ Cn

and

Y`+1 = σn(Y`), ` ∈ N.

The idea is that kn(Y`) is a good approximation for the number of points in the F -orbit of

Y lying inside the slice [Y`, Y`+1). Hence,

Claim 5.4.4. For k ∈ N, set tk = Φ(k)(Y ) and `k = `Cn(tk, Y ) = cardt ∈ [0, tk) | Y − t ∈Cn (as defined in Lemma 5.2.2). Let j0 = minj ∈ Z | F j(Y ) Y0. Then, for every k ∈ Ngreater or equal than j0 we have:

(5.4.3) j0 +

`k−2∑`=0

kn(Y`) ≤ k ≤ j0 +

`k−1∑`=0

kn(Y`) + `k.

Proof. For each ` ∈ N \ 0 define j` to be the unique integer satisfying

(5.4.4) F j`(Y ) ∈ [Y`, F (Y`)).

Observe that by definition, j0 satisfies (5.4.4) and that the existence and the uniqueness of

j` follow from the facts that F is orientation-preserving and Y belongs to Ω+X .

The difference j`+1 − j` is exactly the number of iterations that the orbit of Y by F lies

inside the slice [Y`, Y`+1). We show that either

(5.4.5) j`+1 = j` + kn(Y`) or j`+1 = j` + kn(Y`) + 1.

Indeed, since F is orientation-preserving, iterating (5.4.4) by F kn(Y`) times yields that

F j`+kn(Y`)(Y ) belongs to [F kn(Y`)(Y`), Fkn(Y`)+1(Y`)]. By the definition of kn(Y`), this slice

72


Figure 5.4: Proof of Theorem 5.4.3: On the left: j`+1 = j` + kn(Y`). On the right: j`+1 =j` + kn(Y`) + 1.

clearly intersects [Y`+1, F (Y`+1)). Now we distinguish two cases: (1) F j`+1(Y ) belongs to

the intersection of the two slices (see the left part of Figure 5.4), in which case the equality

j`+1 = j` + kn(Y`) follows from the uniqueness of j`; (2) F j`+1(Y ) F kn(Y`)+1(Y`) in which

case we check j`+1 = j` + kn(Y`) + 1 using the uniqueness of j` (see the right part of Figure

5.4).

Applying (5.4.5) recursively yields to

(5.4.6) j` = j0 +`−1∑`=0

kn(Y`) + e(`)

for every ` ∈ N, with an error e(`) that satisfies 0 ≤ e(`) ≤ `.

On the other hand, we check that F k(Y ) ∈ [Y`k , Y`k+1]. Thus, (5.4.4) and the preservation

of orientation of F imply that

(5.4.7) j`k−1 ≤ k ≤ j`k .

To conclude the proof of the claim we use (5.4.6) to replace j`k and j`k−1 in (5.4.7) which

yields (5.4.3).

We divide (5.4.3) by Φ(k)(Y ) to obtain

(5.4.8)j0

tk+`k − 1

tk

1

`k − 1

`k−2∑`=0

kn(Y`) ≤k

tk≤ j0

tk+`ktk

1

`k

`k−1∑`=0

kn(Y`) +`ktk.

And then we let k go to infinity. Applying Lemma 5.2.2 and Birkhoff’s Ergodic Theorem to

the limit of (5.4.8) yields that, for µ-almost every Y ∈ Ω+X ,

(5.4.9) lim supk→+∞

k

Φ(k)(Y )− ν(Cn) ≤

∫Cn

kndν ≤ lim infk→+∞

k

Φ(k)(Y ).

73


Now we let n go to infinity in (5.4.9). Since diam(Cn)→ +∞, it follows that ν(Cn)→ 0 and

hence we obtain

(5.4.10) lim supn→+∞

∫Cn


k

Φ(k)(Y )≤ lim sup

k→+∞

k

Φ(k)(Y )≤ lim inf

n→+∞

∫Cn

kndν

for µ-almost every Y in Ω+X . Now take Y ∈ Ω+

X such that (5.4.10) holds for Y . From (5.4.10)

it is easy to check that (k/tk)k∈N and (∫Cnkndν)n∈N have limits in [0,+∞) and the are equal.

Denote by α this limit. Since tk ≤ ‖Φ‖∞k for all k, we check that α is greater than 0. Hence

ρ(F, Y ) = limn→+∞

1∫Cnkndν

,

where the latter limit is equal to 0 if α = +∞, and this concludes the proof.

Before concentrating in the case without fixed points, we remark that Theorem 5.4.3 implies

that the set

A = Y ∈ ΩX | ρ(F, Y ) = ρ

satisfies µ(A) = 1. Let µF be an ergodic F -invariant probability measure. Since A is F -

invariant, ergodicity implies that either µF (A) = 0 or µF (A) = 1. If µF (A) = 1 then by

Birkhoff’s Ergodic Theorem there is a point Y ∈ A for which the translation number at Y

coincides with∫

ΩXφdµF and therefore

∫φdµF = ρ. If the case µF (A) = 0 is possible or not

is not addressed in this dissertation. Nevertheless, we believe that this case is not possible

which would bring us to the following refinement of Theorem 5.4.3 (which is analogous to

Theorem 1 in [Kwapisz 00]):

Conjecture 5.4.5. Let F be a map satisfying the hypothesis of Theorem 5.4.3. Then, there

exists ρ ≥ 0 such that for every ergodic F -invariant measure µF supported on Ω+X we have∫

ΩX

ΦdµF = ρ.

Now we concentrate in the case without fixed points. In this case, Ω+X = ΩX and one

would expect that there is a unique translation number for every Delone set in ΩX . In fact,

if we could ensure that the functions kn in the proof of Theorem 5.4.3 are continuous, then a

straightforward modification (by invoking the uniform version of Birkhoff’s Ergodic Theorem

to obtain uniform convergence) would allow us to obtain the desired conclusion. However,

74


kn is not continuous at points where F kn(Y )(Y ) = σn(Y ). Nevertheless, the following lemma

states that the functions kn are not too far from being continuous.

Lemma 5.4.6. Suppose that inf Φ > 0. Let C be a local vertical and let κ : C → N be defined

by

κ(Y ) = maxk ≥ 0 : F k(Y ) σ(Y )

where σ denotes the first return maps to C. Then, there exists κ : C → R which is continuous

and satisfies supY ∈C |κ(Y )− κ(Y )| < 1.

Proof. Let a = inf Φ > 0. We show that κ is bounded. Indeed, Φ(k)(Y ) > ak for every k ∈ Nand Y ∈ C. Since tC is continuous (thus bounded) by Lemma 5.2.1, there exists K > 0 such

that κ(Y ) < K for every Y ∈ C.

The family Φ(k)K+1k=1 is equicontinuous, which means that there is δ > 0 such that for

every k ∈ 1, . . . , K + 1

(5.4.11) d(Y, Z) < δ implies that |Φ(k)(Y )− Φ(k)(Z)| < a

2.

By Lemma 5.2.1, one can find a partition C = CiLi=1 of C by clopen sets with diameter

smaller than δ such that tC restricted to Ci is constant for each i ∈ 1, . . . , L. By definition

of κ we have

(5.4.12) Φ(κ(Z))(Z) ≤ tC(Z) < Φ(κ(Z)+1)(Z)

for all i ∈ 1, . . . , L and all Z ∈ Ci. Fix i ∈ 1, . . . , L. If the first inequality in (5.4.12) is

strict for every Z ∈ Ci, then κ is continuous on Ci and we let κ(Z) = κ(Z) for every Z ∈ Ci.On the contrary, suppose there exists Z∗ ∈ Ci such that Φ(κ(Z∗))(Z∗) = tC(Z∗). Since tC(Z)

is constant over Ci, it follows from (5.4.11) that

(5.4.13) tC(Z)− a

2< Φ(κ(Z∗))(Z) < tC(Z) +

a

2

for every Z ∈ Ci. Hence, combining Φ(κ(Z∗)−1)(Z) ≤ Φ(κ(Z∗))(Z)− a with the first inequality

in (5.4.13) and Φ(κ(Z∗))(Z) + a ≤ Φ(κ(Z∗)+1)(Z) with the last inequality in (5.4.13) we obtain

Φ(κ(Z∗)−1)(Z) ≤ tC(Z)− a

2< tC(Z) < tC(Z) +

a

2≤ Φ(κ(Z∗)+1)(Z).

The latter implies that κ(Z) ∈ κ(Z∗)−1, κ(Z∗) and thus defining κ(Z) = κ(Z∗) for Z ∈ Ciwe get the result.

75


Finally, we are able to state the main result of this chapter establishing the existence of a

unique translation number for maps in F+0 (ΩX) without fixed points. Again, the case Φ < 0

may be treated in an analogous way.

Theorem 5.4.7. Let F be a map in F+0 (ΩX) and Φ be its displacement. Suppose that

inf Φ > 0. Then there exists ρ > 0 such that for every Y ∈ Ω+X , the translation number

ρ(F, Y ) exists and is equal to ρ.

Proof. The proof is a modification of the proof of Theorem 5.4.3. In short, the estimates

in the former proof combined with the Lemma 5.4.6 applied to the kn’s gives estimates

for continuous functions on an uniquely ergodic system. The latter allows to apply the

uniform version of Birkhoff’s Ergodic Theorem to conclude. More precisely, let (Cn)n∈N be a

decreasing sequence of local verticals in ΩX with diam(Cn) → 0 as n → +∞, denote by σn

the first return map to Cn and define kn : Cn → R ∪ +∞ by

kn(Y ) = maxk ≥ 0 : F k(Y ) σn(Y ), Y ∈ Cn

and (Y`)`∈N satisfies

Y0 =

Y − tCn(Y ), if Y 6∈ Cn,

Y, if Y ∈ Cn

and

Y`+1 = σn(Y`), ` ∈ N.

For each n ∈ N, applying Lemma 5.4.6 to kn yields a continuous function kn that satisfies

|kn(Y )− kn(Y )| < 1, for all Y ∈ ΩX .

Combining last inequality with the assertion in Claim 5.4.4 and then dividing by tk yields

(5.4.14)j0

tk+`k − 1

tk

(−1 +

1

`k − 1

`k−2∑`=0

kn(Y`)

)≤ k ≤ j0

tk+`ktk

(1

`k

`k−1∑`=0

kn(Y`) + 1

)+`ktk

for every Y ∈ ΩX (recall that in this case Ω+X = ΩX). We let k go to infinity in (5.4.14). Since

the system (Cn, ν/ν(Cn), σn) is uniquely ergodic, applying the uniform version of Birkhoff’s

Ergodic Theorem and Lemma 5.2.2 we get

(5.4.15) lim supk→+∞

k

Φ(k)(Y )− 2ν(Cn) ≤

∫Cn


k

Φ(k)(Y )+ ν(Cn)

76


for every Y ∈ ΩX . But |∫Cnkn(Y )dν −

∫Cnkn(Y )dν| < ν(Cn) and thus

lim supk→+∞

k

Φ(k)(Y )− 3ν(Cn) ≤

∫Cn


k

Φ(k)(Y )+ 2ν(Cn)

for every Y ∈ Cn. Finally, we take limit when n goes to infinity and proceed exactly as in the

proof of Theorem 5.4.3 to conclude that there is a unique translation number ρ ≥ 0. Since

inf |Φ| > 0, it is easy to check that ρ > 0.

77

6 Towards a Poincare-like classification

for homeomorphisms of Delone

systems homotopic to the identity

Let X be an an aperiodic repetitive Delone set and ΩX its hull. In this chapter we will suppose

that (ΩX ,Γ) is uniquely ergodic and µ will denote the unique Γ-invariant probability measure

on ΩX .

Define H++(ΩX) by

H++(ΩX) = F ∈ F+0 (ΩX) | F is a fixed-point free homeomorphism.

By Theorem 5.4.7 every homeomorphism in H++(ΩX) has a unique translation number. The

objective of this chapter is to study H+ + (ΩX). To motivate the results, we focus first

in the periodic case, where H++(ΩX) corresponds to the set of all orientation-preserving

homeomorphisms of the circle R/Z, for which the following classification was obtained by

Poincare (see [Katok 95] for a proof):

Poincare’s Theorem. Let F : R/Z → R/Z be an orientation-preserving homeomorphism.

Then:

• The rotation number ρ(F ) is rational if and only if F has a periodic orbit.

• If the rotation number ρ(F ) is irrational then F is semiconjugate to Rρ(F ), which means

that there exists a continuous, surjective, orientation-preserving map H : R/Z→ R/Zsuch that

H F = Rρ(F ) H

where Rρ(F ) is the rotation x 7→ x+ ρ(F ) mod Z in R/Z.

78

6 Towards a Poincare’s Theorem

The main objective of this chapter is to obtain a Poincare-like classification of the homeo-

morphisms in H++(ΩX) for an aperiodic X. In this context, rational numbers are no longer

available. Hence, we need to find a good replacement, a set Q (defined in terms of ΩX) that

will play the role of the rational numbers in our results. We do so in Section 6.1. We also

observe that homeomorphisms in H++(ΩX) do not have periodic points. We introduce in

Section 6.1 the notion of local strips, which will play the role of the periodic points.

A key fact used in the proof of Poincare’s Theorem is the following: if F : R/Z→ R/Z is

a orientation preserving homeomorphism then for every t ∈ R we have

(6.0.1) |F n(t)− t− nρ(F )| < 1.

It is an open question whether this is true in the aperiodic case. Thus, motivated by the

recent work of Jager and Stark[Jager 06], we restrict our attention to homeomorphisms of ΩX

for which the cocycle corresponding to (6.0.1) is bounded. We refer to these homeomorphisms

as ρ-bounded and study these maps in Section 6.2. In the last section, we state and prove

our results which give a classification of ρ-bounded homeomorphisms in H++(ΩX) according

to their translation numbers.

6.1 Replacing the rational numbers

In order to state a Poincare-like classification theorem in the context of aperiodic minimal

Delone systems it is necessary to obtain a proper replacement for rational numbers. To

motivate our definition we recall the well known fact that a rotation x 7→ x + ρ mod Z of

the circle R/Z is minimal if and only if ρ is irrational.

Let (ΩX ,Γ) be a minimal Delone system of the real line. We define

Q = t ∈ R : (ΩX ,Γt) is not minimal.

If ρ ∈ Q then we say that ρ is Γ-rational and in the opposite case we say that ρ is

Γ-irrational . There is a well-known description of Q in terms of continuous eigenvalues of

(ΩX ,Γ). First, we recall that λ ∈ R is called a continuous eigenvalue of (ΩX ,Γ) if there

exists a continuous function vλ : ΩX → S1 such that

(6.1.1) vλ(ΓtY ) = e2πiλtvλ(Y ) for all t ∈ R and Y ∈ ΩX ,

79


where S1 = z ∈ C : |z| = 1 denotes the unit circle. The function vλ is called the

continuous eigenfunction associated with λ. It is easy to see that vλ is onto if λ 6= 0.

The following result gives the announced description of Q (an outline of the proof is given in

[Glasner 03, 4.24.1]).

Proposition 6.1.1.

(6.1.2) Q =

k

λ: k ∈ Z, λ 6= 0 is a continuous eigenvalue of (ΩX ,Γ)

.

In particular, Q is countable.

Proof. Let λ 6= 0 be a continuous eigenvalue of (ΩX ,Γ) and k ∈ Z an integer. Let us proceed

by contradiction and suppose that (ΩX ,Γ kλ) is minimal. Consider the set K = v−1

λ (1).We check that K is closed and proper because vλ is continuous and onto (since λ 6= 0).

Applying (6.1.1) with t = kλ

yields vλ(Γ kλY ) = vλ(Y ) for every Y ∈ ΩX . It follows that K is

Γ kλ-invariant. Hence, by minimality of(ΩX ,Γ k

λ) we have either K = ∅ or K = ΩX but any

of these cases contradicts the fact that vλ is onto. Therefore, (ΩX ,Γ kλ) is not minimal and k

λ

belongs to Q.

Now we show the converse inclusion. Let q ∈ Q and let ∅ 6= K ⊂ ΩX be an Γq-invariant

set. We suppose that the system (K,Γq) is minimal and consider

I = t ∈ R |ΓtK = K.

Minimality of (ΩX ,Γ) implies that I is a proper subset of R. Moreover, it is easy to check

that I is a closed subgroup of R. Hence, there exists s0 > 0 such that I = s0Z. This is

well-known but we provide an outline for completeness. Let s0 = infs > 0 | s ∈ I. If we

suppose that s0 = 0, then we check that 0 is an accumulation point of I. It follows that Iis dense in R. But since I is closed, we deduce that I = R, which gives us a contradiction

and therefore s0 > 0 and I ⊇ s0Z. If s ∈ I, then there exist k ∈ Z and |r| < s0 such that

s = ks0 + r. This implies that r ∈ I and hence I ⊆ s0Z.

We consider the family K = ΓsK : s ∈ [0, s0). For each s ∈ [0, s0) the system (ΓsK,Γq)

is minimal. Hence, the sets in K either coincide or do not intersect. Now we check that Kis pairwise disjoint. Indeed, suppose that there exist s, s′ ∈ [0, s0) such that ΓsK = Γs′K.

This implies that |s− s′| ∈ I. Since |s− s′| < s0, it follows that s = s′. On the other hand,

80


minimality of (Ω,Γ) and the fact that K is not empty imply

ΩX =⋃

s∈[0,s0)

ΓsK

and the family K defines a partition of ΩX . This means that each Y ∈ ΩX can be written in

the form Y = Γs(Y )YK , where YK ∈ K and s(Y ) ∈ [0, s0). We check that s(ΓtY ) = t + s(Y )

mod s0 and that the map Y 7→ s(Y ) mod I is continuous. Indeed, if (Yn)n∈N is a sequence

converging to Y ∈ ΩX , then Yn = Yn,K−s(Yn). By the compactness of K and R/I, dropping

to a subsequence we may assume that (Yn,K)n∈N has a limit Z ∈ K and s(Yn) has a limit

s ∈ [0, s0]. From the continuity of Γ, it follows that

Z − s = YK − s(Y )

from which continuity follows. Let v be the continuous function v : ΩX → S1 defined by

v(Y ) = exp

(2πi

s(Y )

s0

).

We check that

v(ΓtY ) = v(Γt+s(Y )YK) = exp

(2πi

t+ s(Y )

s0

)= exp (2πit) v(Y )

for every Y ∈ ΩX . Therefore, λ = 1s0

is a continuous eigenvalue of (ΩX ,Γ). Clearly q ∈ I =

s0Z. Thus, there exists k ∈ Z such that q = kλ

and the description of Q is done. To see

that Q is countable, it suffices to recall the well-known fact that the group of eigenvalues of

a compact dynamical system is countable (see [Walters 82]).

6.2 Γ-semiconjugacies and ρ-bounded maps

We recall that given two homeomorphisms F,G ∈ H++(ΩX), F is semiconjugate to G if

there is a continuous onto map H : ΩX → ΩX such that

(6.2.1) H F = G H.

The map H is referred to as the semiconjugacy from F to G and when H is also one-to-one

then F and G are said to be conjugate and the map H is called a conjugacy. The natural

81


question we address here is whether translation numbers are preserved under semiconjugacy.

The following example show that this is not always the case:

Example 6.1. Let X = Fib be the Fibonacci Delone set defined in Example 5.1. The

inflation-substitution mapping extends to a homeomorphism ω on ΩX that fixesX. Moreover,

it is easy to check that

(6.2.2) ω(Y − t) = ω(Y )− τt

for every Y ∈ ΩX and t ∈ R, where τ denotes the golden-ratio. Let F be a homeomorphism

in H++(ΩX) and Φ its displacement. We suppose that Φ is strictly positive and we define

ΦG by

ΦG(Y ) = λΦ(ω−1(Y )) for every Y ∈ ΩX .

If we define G : ΩX → ΩX by G(Y ) = Y − ΦG(Y ), then we check

G(ω(Y − t)) = ω(Y − t)− τΦ(Y − t) =Y − τ(t + Φ(Y − t))

=ω(Y − t− Φ(Y − t)) = ω(F (Y − t))

for every t ∈ R. Since F and ω are continuous map, a density argument yields that F and

G are conjugate by ω. It can be checked that G preserves orientation. Hence G belongs

to H++(ΩX) and it has a translation number by Theorem 5.4.7. We now compute ρ(G).

Iterating the semiconjugacy equation (6.2.1) gives

Gn(Y ) = ω F n ω−1(Y ) for all Y ∈ ΩX and n ∈ N.

This equality applied to X reads

X − Φ(n)G (X) = ω(X)− Φ(n)(X) = X − τΦ(n)(X).

The aperiodicity of X then implies that

ρ(G) = τρ(F ).

We observe that the ω is not homotopic to the identity. Indeed, this easily follows from

Theorem 5.3.3 and (6.2.2).

Motivated by this, we restrict our attention to semiconjugacies that are also homotopic

82


to the identity: we say that F ∈ H++(ΩX) is Γ-semiconjugate to G ∈ H++(ΩX) if there

exists a semiconjugacy H from F to G that is homotopic to the identity. In this case, we call

H a Γ-semiconjugacy from F to G. The proof of the following result is a direct modification

of the proof of the fact that rotation numbers are preserved by semiconjugacies and is left to

the reader (see [Katok 95]).

Proposition 6.2.1. If F ∈ H++(ΩX) is Γ-semiconjugate to G ∈ H++(ΩX) then the trans-

lation numbers of F and G coincide.

Let F ∈ H++(ΩX) and ρ(F ) its translation number. The natural question to solve in

a Poincare-like classification is whether F is Γ-semiconjugate to Γρ(F ). Recall that Γρ(F ) is

defined by Y 7→ Y − ρ(F ). To give the following definition, we first suppose that there is a

Γ-semiconjugacy H from F to Γρ(F ). In such case, if we write the semiconjugacy equation

(6.2.1) in terms of the displacements of F and H we obtain the following cohomological

equation:

(6.2.3) Ψ(F (Y ))−Ψ(Y ) = ρ(F )− Φ(Y ), Y ∈ ΩX ,

where Φ and Ψ denote, respectively, the displacements of F and H. Thus, the problem

of finding a Γ-semiconjugacy reduces to the problem of finding a continuous (and therefore

bounded) solution to (6.2.3). A natural necessary condition to have bounded solutions for

cohomological equations is that the associated cocycle:

ζF (n, Y ) = nρ(F )− Φ(n)(Y )

to be bounded. Indeed, by applying (6.2.3) recursively we obtain

Ψ(F n(Y ))−Ψ(Y ) = nρ(F )− Φ(n)(Y )

from which it follows that |ζF (n, Y )| < 2‖Ψ‖∞. In the periodic case, this condition is always

satisfied. However, for a map F ∈ H++(ΩX), it is not known to the author whether the

associated cocyle is bounded or not. Hence, we say that F is ρ-bounded if there exists

C > 0 such that the cocycle

|ζF (n, Y )| < C for all Y ∈ ΩX and n ∈ N.

Thus, ρ-boundedness is a necessary condition for F to be Γ-semiconjugate to Γρ(F ). When

F is minimal, then this condition is also sufficient by the following theorem due to Gottschalk

83


and Hedlund (see [Katok 95, Section 2.9] for an introduction to cohomological equations and

a proof of this theorem):

Theorem 6.2.2 (Gottschalk-Hedlund). Let M be a compact metric space and T : M → M

be a minimal homeomorphism. Let φ be a continuous function over M and suppose there is

C > 0 such that

|k−1∑i=0

φ(T i(m))| < C

for every k ∈ N and m ∈M . Then the cohomological equation

ψ(T (m))− ψ(m) = φ(m), m ∈M

has a continuous solution.

In the following, we will suppose that F is ρ-bounded. An easy consequence of Theorem

6.2.2 is the following:

Corollary 6.2.3. If F ∈ H++(ΩX) is minimal and ρ-bounded, then F is Γ-semiconjugate to

Γρ(F ) and ρ(F ) is Γ-irrational.

This implies that at least in the minimal case, Γ-rational numbers provide a good replace-

ment of rational numbers. Next result gathers the consequences of Theorem 6.2.2 in our

context without assuming the minimality of F .

Lemma 6.2.4. Suppose that F ∈ H+(ΩX) is ρ-bounded. Then the equation (6.2.3) has a

bounded solution Ψ : ΩX → R. Moreover, its restriction to any F -minimal set is continuous

and the map H defined by H(Y ) = Y −Ψ(Y ) for Y ∈ ΩX is orientation-preserving and the

following diagram commutes:

(6.2.4)

ΩXF−−−→ ΩX

H

y yHΩX

Tρ(F )−−−→ ΩX

Proof. We let

Ψ(Y ) := lim supn∈N

Φ(n)(Y )− nρ(F ).

84


Ψ is well defined and bounded because F is ρ-bounded and it solves the cohomological

equation (6.2.3) (for details see [Katok 95, Theorem 2.9.3]). Let K be a F -minimal set. By

following the proof of Gottschalk-Headlund’s Theorem given in [Katok 95, Theorem 2.9.4] we

check that the restriction of Ψ to K is continuous. To see that H is orientation preserving,

it suffices to prove that the function hY defined by t ∈ R 7→ t + ψ(Y − t) is non-decreasing

for every Y ∈ ΩX . Indeed,

t + Ψ(Y − t) = lim supn∈N

t + Φ(n)(Y − t)− nρ(F )

for every t ∈ R. Besides, t 7→ t + Φ(n)(Y − t) is non-decreasing because F n preserves

orientation and F n(Y ) = Y − Φ(n)(Y ). As the lim sup of non-decreasing functions is non-

decreasing the proof is finished.

6.3 Towards a Poincare Theorem for ρ-bounded

homeomorphisms in H++(ΩX)

Let F be a homeomorphism inH++(ΩX) and Φ its displacement. To fix ideas, we assume that

inf Φ > 0. We will suppose that F is ρ-bounded and we let Ψ and H be as in Lemma 6.2.4.

That is, Ψ is a bounded solution to the cohomological equation (6.2.3) and H : ΩX → ΩX is

defined by H(Y ) = Y − Ψ(Y ) for every Y ∈ ΩX . To obtain our Poincare-like classification

theorem, first in the Γ-rational case we will use H to describe minimal sets and then in

the Γ-irrational case we will prove that H is continuous and onto and thus defines a Γ-

semiconjugacy from F to Γρ(F ). We state here some basic lemmas about F -invariant sets

that will needed in both cases are do not depend on the Γ-rationality of ρ(F ).

Lemma 6.3.1. Let K be a non-empty closed F -invariant set. Then K intersects every leaf

of ΩX . Moreover, the set of return-times RK(Y ) is closed and relatively dense in R for every

Y ∈ ΩX .

Proof. Let Y ∈ ΩX . The set of return-times RK(Y ) is the preimage of K by the continuous

map t 7→ Γt(Y ) and hence it is closed.

Now we prove that if RK(Y ) is not empty, then it is relatively dense. Indeed, suppose that

RK(Y ) is not empty. There is t ∈ R such that Y −t ∈ K and by F -invariance of K it follows

that the F -orbit of Y − t is included in K. Thus, the set of return-times contains the set

85


t + Φ(n)(Y − t) : n ∈ Z . But the latter set is relatively dense because Φ(n)(Y ) : n ∈ Zis unbounded (from above and below) and its gaps are smaller than sup Φ, which is finite

because Φ is continuous. This implies that RK(Y ) is relatively dense.

To conclude the proof of the Lemma, it suffices us to show that RK(Y ) is not empty for

every Y ∈ ΩX . Fix Y ∈ K. By definition, RK(Y ) is not empty. Applying the previous

argument to Y we obtain that RK(Y ) is relatively dense. Since (Ω,Γ) is minimal, it follows

that for every Z ∈ ΩX there is a sequence (tn)n∈N ⊆ R such that (Y − tn)n∈N converges to

Z. The relative density of RK(Y ) implies that for each n ∈ N there is sn ∈ RK(Y ) such

that the sequence (sn − tn)n∈N is bounded. Hence, by dropping to a subsequence we may

suppose that (sn − tn)n∈N converges to some s ∈ R. It follows that (Y − sn)n∈N converges

to Z − s. But by construction, the sequence (Y − sn)n∈N is included in K, which is closed.

Hence, Z − s belongs to K which means that s ∈ RK(Y ) and the proof is done.

Lemma 6.3.2. Let K be a non-empty F -minimal set. Then H(K) is closed and Tρ(F )-

minimal.

Proof. By Lemma 6.2.4, the map H|K is continuous. Since K is compact, it follows that

H(K) is compact and thus closed. The fact that H(K) is Γρ(F )-invariant follows from (6.2.4).

Finally, suppose that L is a closed Γρ(F )-invariant subset of H(K). Then, H−1(L) ∩K is a

non-empty closed F -invariant set. From the minimality of the system (K,L), it follows that

H−1(L) ⊇ K and thus L ⊇ H(K), which concludes the proof.

6.3.1 The Γ-rational case

In this section we consider the case where ρ(F ) is a Γ-rational number, and we give a de-

scription of F -minimal sets. We start by describing F -minimal sets in the simplest case, i.e.,

when F = Γρ is a translation.

Proposition 6.3.3. For ρ ∈ Q, the Γρ-minimal sets are transversals.

Proof. Let ρ ∈ Q. By (6.1.2), there is non-zero k ∈ Z such that λ = kρ

is a continuous

eigenvalue of (Ω,Γ). Denote by vλ the continuous eigenfunction associated to λ. For each

θ ∈ S1, we define Iθ = v−1λ (θ). Clearly, Iθ is Γ ρ

k-invariant for each θ ∈ S1. Lemma 6.3.1

implies that Iθ intersects every Γ-orbit. By (6.1.1), we check that for a given Y ∈ Iθ, the

Delone set Y − t belongs to Iθ if and only if t ∈ ρkZ. Hence, for every small enough box

86


Ix will meet each slice of the box at most in one point. Moreover, Let Y ∈ Iθ and ε > 0.

There is 0 < δ < ε such that for every Z ∈ ΩX that satifies d(Y, Z) < δ we have that

d(vλ(Y ), vλ(Z)) < ε. We consider V = CY,δ−1 , and define ϕ : V → R by:

ϕ(Z) =ρ

kargvλ(Z)(vλ(Y ))−1, Y ∈ V.

The function ϕ is continuous and the following equation

vλ(Y − ϕ(Y )) = vλ(Y ), Y ∈ V,

implies that Y − ϕ(Y ) ∈ Ix for every Y ∈ V . Thus, V [ϕ] ⊆ Ix and moreover Ix ∩ V [−ε, ε] =

V [ϕ] because supY ∈V |ϕ(Y )| < ε by the choice of δ. Choosing ε small enough one sees that

V [−ε, ε] is a box and therefore Ix is a transversal.

The rest of the proof is divided in three parts: first we will show that the sets Iθ are

Γρ-minimal; then, we check that vλ is Γρ-invariant and thus constant over every Γρ-minimal

set I. This implies that I = Iθ for some θ ∈ S1. Finally, we will prove that for each Y ∈ θ,the set Iθ is a transversal.

Let θ ∈ S1. The proof of the fact that Iθ is Γρ-minimal is standard. First, we check that

Iθ is Γ ρk-minimal. By (6.1.1), Iθ is clearly Γρ/k-invariant, and by the continuity of vλ, it is

also closed. By definition of the Iθ, the family ΓtIθ : t ∈ [0, ρk) defines a partition of ΩX .

Let I 6= ∅ be a Γ ρk-minimal subset of Iθ. We check that the family ΓtI : t ∈ [0, ρ/k) also

defines a partition of ΩX . It is now clear that I = Iθ. To see that Iθ is in fact Γρ-minimal,

we use a well-known argument (see [de Vries 93, II.9.6(7)]). Let I ⊆ Iθ be a Γρ-minimal set.

The set I ∪ Γ 1kρI . . . ∪ Γ k−1

kρI is clearly closed, Γ ρ

k-invariant. Hence, from the Γ k

ρ-minimality

of Iθ, it follows that Iθ = I ∪ Γ 1kρI . . . ∪ Γ k−1

kρI. For each l ∈ 1, . . . , k − 1, the set Γ l

kρI is

Γρ-minimal. Thus, either Γ lkρI ∩ I = ∅ or Γ l

kρI = I. Let l′ be the smallest l′ ≥ 1 such that

Γ l′kρI = I. We check that

Iθ

[l

kρ,l + 1

kρ

): l ∈ 0, . . . , l′ − 1

is a finite clopen partition of ΩX . Since ΩX is connected, it follows that l′ = 1 and thus

I = Iθ.

Let us come back to the general Γ-rational case. By using Lemma 6.2.4 and Proposition

6.3.3 we obtain some local invariant graphs (recall that a local graph is not necessarily closed).

87


Since H may not be continuous, these invariant graphs are not closed. This motivates us to

study the closures of local graphs, for which we need we need the following definitions: for a

local vertical V in ΩX , we let RV = 12

infY ∈V tV (Y ) > 0. Given two functions α, β : V → Rsatisfying −RV < α ≤ β < RV , the set

V [α, β] = Y − t : Y ∈ V, α(Y ) ≤ t ≤ β(Y )

is called a local strip. When Y ∈ V , then Y [α, β] is defined by

Y [α, β] = Y − t | α(Y ) ≤ t ≤ β(Y ).

We check that, by the definition of RV , the restriction of Γ to V × [α, β] is a homeomorphism

from V × [α, β] to V [α, β]. We also remark that V [α] = V [α, α]. A set K is called a strip

if it can be decomposed as a finite disjoint union of local strips. We recall that a function

α : V → R is called lower semicontinuous (abbreviated l.s.c.) if for every Y ∈ V , we

have

lim infYn→Y

α(Yn) ≥ α(Y ).

The function α is said to be upper semicontinuous (abbreviated u.s.c.) if −α is lower

semicontinuous. A local strip V [α, β] is thin when V [α] = V [β] = V [α, β]. A strip is thin

if its the finite union of disjoints thin local strips.

The following results follow directly from the fact that V [α, β] is homeomorphic to V ×[α, β]

and standards results in the theory of semicontinuous functions.

Lemma 6.3.4. A local strip V [α, β] is closed (as a subset of ΩX) if and only if α is lower

semicontinuous and β is upper semicontinuous.

Proposition 6.3.5. The closure of a local graph V [α] is a thin local strip of the form

V [α−, α+] where

α+(x) = infβ(x) : β is upper semicontinuous and β ≥ α on V

and

α−(x) = supβ(x) : β is lower semicontinuous and β ≤ α on V .

Now, we can state the main result of this section:

Theorem 6.3.6. Let F be a ρ-bounded homeomorphism in H++(ΩX). Then, every F -

minimal set is a thin strip.

88


Before giving the proof, we comment that it basically amounts to apply Lemma 6.2.4 to

obtain local invariant graphs by taking the preimage via H of the local transversals given

by Proposition 6.3.3. Then we apply Proposition 6.3.5 to obtain closed invariant strips. To

make this precise, we need the following lemma:

Lemma 6.3.7. Given a local transversal V [ϕ] ⊆ ΩX , there are bounded real-valued functions

t, s over V such that

V [s, t) ⊆ H−1(V [ϕ]) ⊆ V [s, t].

Moreover, if ‖ϕ‖∞ + ‖Ψ‖∞ < RV , then the set V [s, t] is a local strip.

Proof. Since H preserves every leaf of ΩX , it follows that for every Y ∈ V , the set Acontaining the preimages of Y − ϕ(Y ) by H is included in the leaf of Y . This gives a

correspondence between the elements of A and the solutions of the following equation:

(6.3.1) t + Ψ(Y − t) = ϕ(Y ).

Since H preserves the orientation, the function t ∈ R 7→ hY (t) = t + Ψ(Y − t) is non-

decreasing. Moreover, it is unbounded from above and below, since Ψ is bounded by Lemma

6.2.4. Hence, the set of solutions of (6.3.1) is a bounded interval. We denote by s(Y ) and

t(Y ) its lower and upper extreme points, respectively. We observe that every solution t of

(6.3.1) satisfies |t| ≤ ‖Ψ‖∞ + ‖Φ‖∞. This implies that t and s are bounded and also that

V [s, t] is a local strip when ‖Ψ‖∞ + ‖ϕ‖∞ < RV .

Proof of Theorem 6.3.6. Let K be a F -minimal set and set I = H(K). By Lemma 6.3.2,

we know that I is Γρ(F )-minimal. Proposition 6.3.3 implies that I is the finite disjoint union

of local transversals, say V1[ϕ1], . . . , Vn[ϕn]. Without loss of generality, we may assume that

RVi > ‖Ψ‖∞ + ‖ϕi‖∞ (it suffices to take local vertical with small diameter). For each

i ∈ 1, . . . , n, we define

Ki := H|−1K (Vi[ϕi]) = H−1(Vi[ϕi]) ∩K.

Clearly, the family Ki : i ∈ 1, . . . , n defines a finite partition of K. For each i ∈1, . . . , n, we check that Ki is closed in K, since H|K is continuous. It follows that Ki is

closed. Applying Lemma 6.3.7 we obtain real-valued functions si, ti over Vi such that

(6.3.2) Vi[si, ti) ∩K ⊆ Ki ⊆ Vi[si, ti] ∩K for each i ∈ 1, . . . , n.

89


For each i ∈ 1, . . . , n, we define

αi(Y ) = min([si(Y ), ti(Y )] ∩RK(Y ))

and

βi(Y ) = max([si(Y ), ti(Y )] ∩RK(Y )).

From (6.3.2) it easily follows that

(6.3.3) Ki = Vi[αi, βi] ∩K for each i ∈ 1, . . . , n.

We suppose for now that the set A = ∪iVi[αi] is F -invariant. By (6.3.3) and the definition

of the Ki’s, we see that A is included in K. Hence, the F -minimality of K implies that

the closure of A coincides with K. But A is the finite union of the local graphs Vi[αi],

whose closure are the thin local strips Vi[α−i , α

+i ] by Proposition 6.3.5. Each of these strips is

included in the corresponding Ki. Since the Ki’s are disjoint, it follows that the local strips

Vi[α−i , α

+i ] are also disjoint and thus K is the finite disjoint union of thin local strips, which

means that K is a thin strip.

It remains to show that A is Γ-invariant. Indeed, let i ∈ 1, . . . , n and Y ∈ Vi so Y −αi(Y )

belongs to A. Since I is Γρ(F )-invariant, there is j ∈ 1, . . . , n and Z ∈ Vj such that

(6.3.4) Y − ϕi(Y )− ρ(F ) = Z − ϕj(Z).

We remark that j and Z are uniquely defined by (6.3.4). We will prove that

(6.3.5) Z − αj(Z) = F (Y − αi(Y )),

from which it follows that A is F -invariant, since Z − αj(Z) belongs to A. We divide the

proof of (6.3.5) in two parts: we first observe that (6.3.2) implies that

(6.3.6) Y [si, ti] = H−1(Y − ϕi(Y )) and Z[sj, tj] = H−1(Z − ϕj(Z)),

Combining the second relation in (6.3.6) with the definitions of αj and βj yields

Z[αj, βj] ⊆ H−1(Z − ϕj(Z)).

90


Taking preimage by F of this inclusion and replacing (6.2.4) gives

(6.3.7) F−1(Z[αj, βj]) ⊆ H−1(Z − ϕj(Z) + ρ(F )).

Hence, by (6.3.4) and the first relation in (6.3.6), the last relation implies that

(6.3.8) F−1(Z[αj, βj]) ⊆ Y [si, ti].

Since K is F -invariant, it follows from (6.3.8) and (6.3.2) that Z−αj(Z) belongs to Ki. Now,

we deduce from the definition of αi that F−1(Z − αj(Z)) Y − αi(Y ). But F preserves

orientation so this implies that

(6.3.9) Z − αj(Z) F (Y − αi(Y )).

The converse inequality is obtained in the same way. Combining the first relation in (6.3.6)

with the definitions of αi and βi we obtain

(6.3.10) Y [αi, βi] ⊆ H−1(Y − ϕi(Y )).

We apply F to (6.3.10) and then use (6.3.4) and (6.2.4) in the resulting relation to obtain

F (Y [αi, βi]) ⊆ H−1(Z − ϕj(Z)).

Thus, applying the second relation of (6.3.6) to the last inclusion yields

(6.3.11) F (Y [αi, βi]) ⊆ Z[sj, tj].

Since K is F -invariant, it follows from (6.3.11) and (6.3.2) that F (Y −αi(Y )) belongs to Kj.

The definition of αj then implies that F (Y − αi(Y )) Z − αj(Z), and since F preserves

orientation, this implies that

(6.3.12) Z − αj(Z) F (Y − αi(Y )).

91


6.3.2 The Γ-irrational case

In this section, we suppose that ρ(F ) is a Γ-irrational number and complete our Poincare-like

classification. To this, we will prove that the map H is continuous and onto. Therefore, we

have:

Theorem 6.3.8. Let F be a ρ-bounded homeomorphism in H++(ΩX) and Φ its displacement.

If ρ(F ) is Γ-irrational, then F is Γ-semiconjugate to Γρ(F ).

For the proof we need the following definitions: let K be a non-empty closed F -invariant

set. For each Y ∈ ΩX , we define

K+(Y ) = supt ≤ 0 : t ∈ RK(Y )

and

K−(Y ) = inft ≥ 0 : t ∈ RK(Y ).

It is easy to check that

(6.3.13) −‖Φ‖∞ ≤ K+(Y ) ≤ 0 ≤ K−(Y ) ≤ ‖Φ‖∞.

Moreover, the following lemma states that the slice Y [K−, K+] is included in a level-set of

H:

Lemma 6.3.9. Let K be a non-empty closed F -invariant set. Then, we have that Y −K−(Y ) ∈ K and Y − K+(Y ) ∈ K for every Y ∈ ΩX , and the maps Y 7→ K+(Y ) and

Y 7→ K−(Y ) are, respectively, upper semicontinuous and lower semicontinuous. Moreover,

the map H restricted to K is onto and the following equation

(6.3.14) H(Y −K+(Y )) = H(Y ) = H(Y −K−(Y ))

holds for every Y ∈ ΩX .

Proof. We only check the assertions for K+. The assertions for K− can be obtained used

the same arguments. Let Y ∈ ΩX . We show that Y −K+(Y ) belongs to K. Indeed, Lemma

6.3.1 implies that the set AY = (−∞, 0] ∩ RK(Y ) is non-empty and closed and bounded

from above. Hence, it attains it maximum and, since K+(Y ) = supAY we conclude that

K+(Y ) ∈ RK(Y ), which by definition of RK(Y ) means that Y −K+(Y ) ∈ K.

92


Now we show that Y 7→ K+(Y ) is upper semicontinuous. Since K+(Y ) ≤ 0 for every

Y ∈ ΩX , it follows that y defined as lim supZ→Y K+(Z) is finite. Consider a sequence

(Yn)n∈N that converges to Y as n → +∞. Without loss of generality, we suppose that

K+(Yn) converges to y as n → +∞. By the continuity of Γ we have that Yn − K+(Yn)

converges to Y − y. Since this sequence is included in K and K is closed, we deduce that

y ∈ AY . The definition of K+ then implies that y ≤ K+(Y ). This proves that K+ is upper

semicontinuous.

We now show that H restricted to K is onto, which means that H(K) = ΩX . Indeed,

H(K) is Tρ(F )-invariant. Since ρ(F ) is Γ-irrational, we know that (ΩX ,Γρ(F )) is minimal

and hence H(K) = ΩX .

Finally, we suppose by contradiction that (6.3.14) does not hold. This means that there

exists Y ∈ ΩX for which we have H(Y −K+(Y )) 6= H(Y −K+(Y )). Since H preserves the

orientation, we have that H(Y −K+(Y )) H(Y −K−(Y )) and thus there exists Z ∈ ΩX

such that

H(Y −K+(Y )) ≺ Z ≺ H(Y −K−(Y )).

The fact that each leaf is preserved by H implies that for every Y ′ ∈ H−1(Z) ∩ K, which

is not empty by the previous paragraph, there is t ∈ R such that Y ′ = Y − t. Finally, the

preservation of orientation of H implies that t belongs to (K+(Y ), K−(Y )) , which clearly

contradicts either the definition of K+ or the definition of K−.

Proof of Theorem 6.3.8. We just need to prove that H is continuous, since H satisfies (6.2.4)

and is onto by Lemma 6.3.9. Take an arbitrary Y ∈ ΩX and consider a sequence (Yn)n∈N ⊆ΩX that converges to Y . Since Ψ is bounded, by dropping to a subsequence, we have that the

sequence (H(Yn))n∈N converges to some Z ∈ ΩX . By a standard argument, it is sufficient to

prove that Z = H(Y ). To do this, we consider a F -minimal set K. By (6.3.13), the sequence

(K+(Yn))n∈N is bounded by ‖Φ‖∞. Thus, by dropping to a new subsequence, we have that

K+(Yn) → x+ ∈ [−‖Φ‖∞, ‖Φ‖∞] as k → ∞. The key step consists in applying (6.3.14) to

each Yn. This yields

H(Yn) = H(Yn −K+(Yn)) for every k ∈ N.

By taking limit in this equation when k → +∞ we obtain

Z = H(Y − x+).

93


From the upper-semicontinuity of K+, we know that x+ ≤ K+(Y ), which means that Y −x+ Y −K+(Y ). On the other hand, (6.3.14) applied to Y reads

H(Y ) = H(Y −K+(Y )),

and since H preserves orientation, it follows that Z H(Y ). To see that Z H(Y )

and thus, that H is continuous, we apply almost the exact same argument to the sequence

(K−(Yn))n∈N. The only difference being the fact that we use the lower-semicontinuity of K−.

Lemma 6.3.9 also allows us to prove that (ΩX , F ) has a unique minimal set as the following

result shows:

Proposition 6.3.10. Suppose that F satisfies the hypotheses of Theorem 6.3.8. Then, the

set KH defined as the closure of

AH = Y ∈ ΩX : H−1(H(Y )) = Y

is the unique minimal set of the system (ΩX , F ).

Proof. We first show that AH is included in every F -minimal setK. Indeed, take Y ∈ ΩX that

does not belong to K. Then by definition of K+ and K− we have that K+(Y ) < 0 < K−(Y ).

Hence, (6.3.14) implies that H−1(H(Y )) is not a singleton, i.e., Y does not belong to AH .

This means that AH is included in K.

Let us check that AH is F -invariant. Consider Y ∈ AH , suppose that Z ∈ ΩX satisfies

H(Z) = H(F (Y )) and let W ∈ F−1(Z). By (6.2.4) one has that

Γρ(F )(H(W )) = H(F (W )) = H(Z) = H(F (Y )) = Γρ(F )(H(Y )).

By aperiodicity, it follows that H(W ) = H(Y ). The fact that Y belongs to AH implies that

W = Y . Hence Z = F (Y ) and the set AH is F -invariant.

Now we show that AH non-empty. We recall that the function hY : R → R induced

by H in the leaf of Y is defined by hY (t) = t + Ψ(Y − t), and for each t ∈ R, we let

t(t) = maxs ≥ t : hY (t) = hY (s) and s(t) = mins ≥ t : hY (t) = hY (s). It is easy

to check that Y [s(t), t(t)] = H−1(H(Y − t)). Whenever s(t) is not equal to t(t), the slice

94


Y [s(t), t(t)] is a plateau of H in the leaf of Y . By observing that s(t′) = s(t) and t(t′) = t(t)

for every t′ ∈ [s(t), t(t)] we deduce that there is a countable number of plateux of H in the

leaf of Y (each of such intervals is indexed by a rational number). This implies that the

image of H restricted to the plateaux of H in the leaf of Y is countable and hence, it does

not completely cover the leaf. Since H is onto, it follows that AH is non-empty.

To conclude, the fact that AH is F -invariant and non-empty implies that AH is F -invariant

and non-empty. Let K be any F -minimal set. Since AH is included in K, it follows that

AH = K, which proves the uniqueness of a F -minimal set.

6.4 Appendix: a tractable condition for ρ-boundedness

In the practice, we will not be able to compute the exact translation numbers for most

homeomorphisms in H++(ΩX). Therefore, it would be convenient to a way to check for

ρ-boundedness without requiring the computation of the translation number of the given

homeomorphism. A first answer is given by the following result, which is well-known in the

case of homeomorphisms of the torus. The proof we give is an easy modification of the proof

in the torus-case that appeared in the notes given by Beguin in a Summer School in Grenoble.

Since the notes are no easy available, we include the proof for completeness.

Proposition 6.4.1. Let F ∈ H++(ΩX). Then the following are equivalent:

(i) F is ρ-bounded.

(ii) There exist Y0 ∈ ΩX and C > 0 such that |Φ(n)(Y0 − t) − Φ(n)(Y0 − s)| < C for every

t, s ∈ R and n ∈ N.

Proof. The implication (i) ⇒ (ii) follows directly from the definition of ρ-boundedness. We

prove (ii) ⇒ (i) by contradiction. Let Y0 ∈ ΩX and C > 0 such that (ii) holds and suppose

there is n0 ∈ N such that

|Φ(n0)(Y0)− n0ρ(F )| > C.

To fix the ideas we suppose

Φ(n0)(Y0) = n0ρ(F ) + C + η,

95


where η > 0 (the other case is analogous). Then, by (ii) we have

Φ(n0)(Y0 − t) ≥ n0ρ(F ) + η

for all t ∈ R. Let n ∈ N and denote by qn and mn the quotient and the rest of the Euclidean

division of n by n0, respectively. Then, using the fact that Φ(n)(Y0) is a cocycle we obtain

Φ(n)(Y0) =

qn−1∑i=0

Φ(n0)(F in0(Y0)) + Φ(mn)(F qnn0(Y0))

≥qn−1∑i=0

(n0ρ(F ) + η) + a

= qnn0ρ(F ) + qnη + a

≥ nρ(F )− n0ρ(F ) +n− n0

n0

η + a

where a = mini=1...n0−1 infY ∈ΩX Φ(i)(Y ) > 0. Taking the limit when n → +∞ implies that

limn→+∞1nΦn(Y0) > ρ(F ), which clearly gives a contradiction.

96

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100

Index

S-patch, 9

X-equivariant, 16, 63

strongly, 16, 63

Γ-invariant set, 24

Γ-irrational, 79

Γ-rational, 79

Γ-semiconjugacy, 19

Γ-semiconjugate, 83

Rd-solenoid, 34

minimal, 35

ρ-bounded, 20, 83

atlas, 33, 34

atlas of lamination

equivalent, 33

box, 33, 39

with base, 40

box decomposition, 40

base, 40

canonical box decomposition, 42

canonical transversal, 29

center, 14

clopen, 30

colored tilings, 26

continuous eigenfunction, 80

continuous eigenvalue, 79

continuous hull, 12

converges fast, 54

cylinder, 29

cylinder set, 12

decorated tiling, 26

Delone set, 8, 21

R-atlas, 23

R-patch of, 22

continuous hull of a, 24

aperiodic, 22

of finite type, 22

period lattice, 22

repetitive, 22

Delone system, 24

aperiodic, 25

minimal, 24

uniquely ergodic, 32

derived tiling, 41, 42

displacement, 15, 17, 67

domino problem, 10

equivalent, 34

external radius, 40

finite local complexity, 25

finite transverse invariant measure, 38

finite type, 9

foliated chart, 33

height, 40

ideal crystal, 22

101

Index

internal radius, 40

invariant, 12

lamination, 33

minimal, 33

linearly repetitive, 11, 23

local graph, 37

local strip, 20, 88

thin, 20, 88

local strips, 79

local transversal, 37

local vertical, 12, 30, 37, 39

lower semicontinuous, 88

measure

Γ-invariant, 31

neighbors, 27

occurrence radius, 21

of a box

base, 40

external radius, 41

internal radius, 41

orbit, 24

parametrization, 39

patch, 9

diameter, 25

return vector of, 22

support, 25

Penrose rhombs, 10

Penrose tilings, 10

prototiles, 26

proximity radius, 21

punctured tile, 26

punctured tiling, 26

quasicrsytal, 62

quasicrystal, 9

range, 16

recognition radius, 40, 41

repetitive, 9, 25

repetitivity function, 23

rotation number of F , 15

semiconjugacy, 81

semiconjugate, 81

set

relatively dense, 21

uniformly discrete, 21

set of entrance times, 30

set of punctures, 27

set of return times, 30

short-range potential, 16, 63

slice, 33, 39

strip, 88

thin, 88

tile, 25

tiles, 9

tiles the plane, 9

tiling, 9, 25

patch, 9

patch of, 25

polyhedral, 25

tiling topology, 23

topological dynamical system, 12

tower system, 45

transition maps, 33

translated, 22

translation invariant, 12

translation number, 62, 69

translation number of f at t, 15

translation number of F at Y ∈ ΩX , 18

translation set, 18, 70

102

Index

transversal, 37

space, 33

uniform pattern frequencies, 12, 31

uniquely ergodic, 12

upper semicontinuous, 88

Van Hove sequence, 31

Voronoi box decomposition, 43

Voronoi cell, 27

Voronoi tiling, 28

Wang system, 9

aperiodic, 10

Wang tile, 9

Wang’s conjecture, 10

zoomed out, 44

103

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