P . I . C . M . – 2018Rio de Janeiro, Vol. 1 (731–758)

DYNAMICAL SYSTEMS, FRACTAL GEOMETRY ANDDIOPHANTINE APPROXIMATIONS

C G T A M

Abstract

We describe in this survey several results relating Fractal Geometry, DynamicalSystems and Diophantine Approximations, including a description of recent resultsrelated to geometrical properties of the classical Markov and Lagrange spectra andgeneralizations in Dynamical Systems and Differential Geometry.

1 Introduction

The theory of Dynamical Systems is concerned with the asymptotic behaviour of sys-tems which evolve over time, and gives models for many phenomena arising from nat-ural sciences, as Meteorology and Celestial Mechanics. The study of a number of thesemodels had a fundamental impact in the development of the mathematical theory of Dy-namical Systems. An important initial stage of the theory of Dynamical Systems wasthe study by Poincaré of the restricted three-body problem in Celestial Mechanics in latenineteenth century, during which he started to consider the qualitative theory of differen-tial equations and proved results which are also basic to Ergodic Theory, as the famousPoincaré’s recurrence lemma. He also discovered during this work the homoclinic be-haviour of certain orbits, which became very important in the study of the dynamicsof a system. The existence of transverse homoclinic points implies that the dynamicsis quite complicated, as remarked already by Poincaré: “Rien n’est plus propre à nousdonner une idée de la complication du problème des trois corps et en général de tous lesproblèmes de Dynamique...”, in his classic Les Méthodes Nouvelles de la MécaniqueCéleste (Poincaré [1987]), written in late 19th century. This fact became clearer muchdecades later, as we will discuss below.

Poincaré’s original work on the subject was awarded a famous prize in honour ofthe 60th birthday of King Oscar II of Sweden. There is an interesting history related tothis prize. In fact, there were two versions of Poincaré’s work presented for the prize,whose corresponding work was supposed to be published in the famous journal ActaMathematica - a mistake was detected by the Swedish mathematician Phragmén in thefirst version. When Poincaré became aware of that, he rewrote the paper, including thequotation above calling the attention to the great complexity of dynamical problems

MSC2010: primary 37D20; secondary 11J06, 28A80, 37C29, 37D40.Keywords: Fractal geometry, Homoclinic bifurcations.

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732 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

related to homoclinic intersections. We refer to the excellent paper Yoccoz [2006] byJean-Christophe Yoccoz describing such an event.

We will focus our discussion of Dynamical Systems on the study of flows (autono-mous ordinary differential equations) and iterations of diffeomorphisms, with emphasisin the second subject (many results for flows are similar to corresponding results fordiffeomorphisms).

The first part of this work is related to the interface between Fractal Geometry andDynamical Systems. We will give particular attention to results related to dynamicalbifurcations, specially homoclinic bifurcations, perhaps the most important mechanismthat creates complicated dynamical systems from simple ones. We will see how thestudy of the fractal geometry of hyperbolic sets has a central rôle in the study of dynam-ical bifurcations. We shall discuss recent results and ongoing works on fractal geometryof hyperbolic sets in arbitrary dimensions.

The second part is devoted to the study of the interface between Fractal Geometryand Diophantine Approximations. The main topic of this section will be the study ofgeometric properties of the classicalMarkov and Lagrange spectra - wewill see how thisstudy is related to the study of sums of regular Cantor sets, a topic which also appearsnaturally in the study of homoclinic bifurcations (which seems, at a first glance, to be avery distant subject from Diophantine Approximations).

The third part is related to the study of natural generalizations of the classical Markovand Lagrange spectra in Dynamical Systems and in Differential Geometry - for instance,we will discuss properties of generalized Markov and Lagrange spectra associated togeodesic flows in manifolds of negative curvature and to other hyperbolic dynamicalsystems. This is a subject with much recent activity, and several ongoing relevant works.

Acknowledgements: We would like to warmly thank Jacob Palis for very valuablediscussions on this work.

2 Fractal Geometry and Dynamical Systems

2.1 Hyperbolic sets and Homoclinic Bifurcations. The notion of hyperbolic sys-temswas introduced by Smale in the sixties, after a global example provided by Anosov,namely the diffeomorphism f (x; y) = (2x + y; x + y) (mod 1) of the torus T 2 =

R2/Z2 and a famous example, given by Smale himself, of a horseshoe, that is a ro-bust example of a dynamical system on the plane with a transverse homoclinic pointas above, which implies a rich dynamics - in particular the existence of infinitely manyperiodic orbits. The figure below depicts a horseshoe.

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 733

(the dynamics sends the square (ABCD) onto the domain bounded by (A0B 0C 0D0)).Let Λ M be a compact subset of a manifold M . We say that Λ is a hyperbolic

set for a diffeomorphism ' : M ! M if '(Λ) = Λ and there is a decompositionTΛM = Es ˚ Eu of the tangent bundle of M over Λ such that D' jEs is uniformlycontracting andD' jEu is uniformly expanding. We say that ' is hyperbolic if the limitset of its dynamics is a hyperbolic set.

It is important to notice that when a diffeomorphism has a transverse homoclinicpoint then its dynamics contains an non-trivial invariant hyperbolic set which is (equiv-alent to) a horseshoe - this explains the complicated situation discovered by Poincaré asmentioned above.

The importance of the notion of hyperbolicity is also related to the stability conjec-ture by Palis and Smale, according to which structurally stable dynamical systems areessentially the hyperbolic ones (i.e. the systems whose limit set is hyperbolic). Afterimportant contributions by Anosov, Smale, Palis, de Melo, Robbin and Robinson, thisconjecture was proved in the C 1 topology by Mañé [1988] for diffeomorphisms and,later, by Hayashi [1997] for flows. The stability conjecture (namely the statement thatstructural stability implies hyperbolicity) is still open in the C k topology for k 2.

As mentioned in the introduction, homoclinic bifurcations are perhaps the most im-portant mechanism that creates complicated dynamical systems from simple ones. Thisphenomenon takes place when an element of a family of dynamics (diffeomorphisms orflows) presents a hyperbolic periodic point whose stable and unstable manifolds havea non-transverse intersection. When we connect, through a family, a dynamics with nohomoclinic points (namely, intersections of stable and unstable manifolds of a hyper-bolic periodic point) to another one with a transverse homoclinic point (a homoclinicpoint where the intersection between the stable and unstable manifolds is transverse) bya family of dynamics, we often go through a homoclinic bifurcation.

Homoclinic bifurcations become important when going beyond the hyperbolic the-ory. In the late sixties, Sheldon Newhouse combined homoclinic bifurcations with thecomplexity already available in the hyperbolic theory and some new concepts in FractalGeometry to obtain dynamical systems far more complicated than the hyperbolic ones.Ultimately this led to his famous result on the coexistence of infinitely many periodic at-tractors (which we will discuss later). Later on, Mora and Viana [1993] proved that any

734 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

surface diffeomorphism presenting a homoclinic tangency can be approximated by a dif-feomorphism exhibiting a Hénon-like strange attractor (and that such diffeomorphismsappear in any typical family going through a homoclinic bifurcation).

Palis conjectured that any diffeomorphism of a surface can be approximated arbitrar-ily well in the C k topology by a hyperbolic diffeomorphism or by a diffeomorphismdisplaying a homoclinic tangency. This was proved by Pujals and Sambarino [2000] inthe C 1 topology. Palis also proposed a general version of this conjecture: any diffeo-morphism (in arbitrary ambient dimension) can be approximated arbitrarily well in theC k topology by a hyperbolic diffeomorphism, by a diffeomorphism displaying a homo-clinic tangency or by a diffeomorphism displaying a heteroclinic cycle (a cycle givenby intersections of stable and unstable manifolds of periodic points of different indexes).A major advance was done by Crovisier and Pujals [2015], who proved that any diffeo-morphism can be approximated arbitrarily well in theC 1 topology by a diffeomorphismdisplaying a homoclinic tangency, by a diffeomorphism displaying a heteroclinic cycleor by an essentially hyperbolic diffeomorphism: a diffeomorphism which displays afinite number of hyperbolic attractors whose union of basins is open and dense.

The first natural problem related to homoclinic bifurcations is the study of homo-clinic explosions on surfaces: We consider one-parameter families ('), 2 (1; 1)

of diffeomorphisms of a surface for which ' is uniformly hyperbolic for < 0, and'0 presents a quadratic homoclinic tangency associated to a hyperbolic periodic point(which may belong to a horseshoe - a compact, locally maximal, hyperbolic invariantset of saddle type). It unfolds for > 0 creating locally two transverse intersectionsbetween the stable and unstable manifolds of (the continuation of) the periodic point. Amain question is: what happens for (most) positive values of ? The following figuredepicts such a situation for = 0.

Fractal sets appear naturally in Dynamical Systems and fractal dimensions when wetry to measure fractals. They are essential to describe most of the main results in thispresentation.

Given a metric space X , it is often true that the minimum number N (r) of balls ofradius r needed to coverX is roughly proportional to 1/rd , for some positive constant d ,when r becomes small. In this case, d will be the box dimension of X . More precisely,

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 735

we define the (upper) box dimension of X as

d (X) = lim supr!0

logN (r)

log r:

The notion ofHausdorff dimension of a set is more subtle, but more useful. The maindifference with the notion of box dimension is that, while the box dimension is relatedto coverings of X by small balls of equal radius, the Hausdorff dimension deals witharbitrary coverings of X by balls of small (but not necessarily equal) radius.

Given a countable covering U of X by balls, U = (B(xi ; ri ))i2N , we define itsnorm jjUjj as jjUjj = maxfri ; i 2 Ng (where ri is the radius of the ball B(xi ; ri )).Given s 2 R+, we define Hs(U) =

Pi2N r

si .

The Hausdorff s-measure of X is

Hs(X) = lim!0

infU covers X

jjUjj<

Hs(U):

One can show that there is an unique real number, the Hausdorff dimension of X ,which we denote by HD(X), such that s < HD(X) ) Hs(X) = +1 and s >HD(X) ) Hs(X) = 0 (soHD(X) can be defined shortly as

HD(X) = inffs > 0; infX[B(xn;rn)

Xrsn = 0g):

For “well-behaved” sets X - in particular for regular Cantor sets in ambient dimen-sion 1 and horseshoes in ambient dimension 2, the box and Hausdorff dimensions of Xcoincide.

Regular Cantor sets on the line play a fundamental role in dynamical systems andnotably also in some problems in number theory. They are defined by expanding mapsand have some kind of self-similarity property: small parts of them are diffeomorphicto big parts with uniformly bounded distortion (we will give a precise definition in awhile). In both settings, dynamics and number theory, a key question is whether thearithmetic difference (see definition below) of two such sets has non-empty interior.

A horseshoe Λ in a surface is locally diffeomorphic to the Cartesian product of tworegular Cantor sets: the so-called stable and unstableCantor setsKs andKu ofΛ, givenby intersections of Λ with local stable and unstable manifolds of some points of thehorseshoe. The Hausdorff dimension of Λ, which is equal to the sum of the Hausdorffdimensions of Ks and Ku, plays a fundamental role in several results on homoclinicbifurcations associated to Λ.

From the dynamics side, in the eighties, Palis Jr. and Takens [1985], Palis Jr. andTakens [1987] proved the following theorem about homoclinic bifurcations associatedto a hyperbolic set:

Theorem 2.1. Let ('), 2 (1; 1) be a family of diffeomorphisms of a surface pre-senting a homoclinic explosion at = 0 associated to a periodic point belonging to ahorseshoe Λ. Assume thatHD(Λ) < 1. Then

limı!0

m(H \ [0; ı])

ı= 1;

whereH := f > 0 j ' is uniformly hyperbolicg.

736 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

2.2 Regular Cantor sets - a conjecture by Palis. A central fact used in the proofof the above theorem by Palis and Takens is that if K1 and K2 are regular Cantor setson the real line such that the sum of their Hausdorff dimensions is smaller than one,then K1 K2 = fx y j x 2 K1; y 2 K2g = ft 2 RjK1 \ (K2 + t) ¤ ¿g (thearithmetic difference between K1 and K2) is a set of zero Lebesgue measure (indeedof Hausdorff dimension smaller than 1). On the occasion, looking for some kind ofcharacterization property for this phenomenon, Palis conjectured (see Palis Jr. [1987],Palis Jr. [1991]) that for generic pairs of regular Cantor sets (K1; K2) of the real lineeither K1 K2 has zero measure or else it contains an interval (the last case shouldcorrespond in homoclinic bifurcations to open sets of tangencies). A slightly strongerstatement is that, if K1 and K2 are generic regular Cantor sets and the sum of theirHausdorff dimensions is bigger than 1, then K1 K2 contains intervals.

Another motivation for the conjecture was Newhouse’s work in the seventies, whenhe introduced the concept of thickness of a regular Cantor set, another fractal invariantassociated to Cantor sets on the real line. It was used in Newhouse [1970] to exhibitopen sets of diffeomorphisms with persistent homoclinic tangencies, therefore with nohyperbolicity. It is possible (Newhouse [1974]) to prove that, under a dissipation hy-pothesis, in such an open set there is a residual set of diffeomorphisms which presentinfinitely many coexisting sinks. In Newhouse [1979], it is proved that under generichypotheses every family of surface diffeomorphisms that unfold a homoclinic tangencygoes through such an open set. It is to be noted that in the case described above withHD(Λ) < 1 (as studied in Palis Jr. and Takens [1987]) these sets have zero density.See Palis Jr. and Takens [1993] for a detailed presentation of these results. An impor-tant related question by Palis is whether the sets of parameter values corresponding toinfinitely many coexisting sinks have typically zero Lebesgue measure.

An earlier and totally independent development had taken place in number theory.In 1947, M. Hall [1947] proved that any real number can be written as the sum of twonumbers whose continued fractions coefficients (of positive index) are bounded by 4.More precisely, if C (4) is the regular Cantor set (see general definition below) formedof such numbers in [0; 1], then one has C (4) + C (4) = [

p2 1; 4(

p2 1)]. We will

discuss generalizations and consequences of this result in the next section.A Cantor set K R is a C k-regular Cantor set, k 1, if:

i) there are disjoint compact intervals I1; I2; : : : ; Ir such thatK I1 [ [Ir andthe boundary of each Ij is contained in K;

ii) there is a C k expanding map defined in a neighbourhood of I1 [ I2 [ [ Irsuch that, for each j , (Ij ) is the convex hull of a finite union of some of theseintervals Is . Moreover, we suppose that satisfies:

ii.1) for each j , 1 j r and n sufficiently big, n(K \ Ij ) = K;

ii.2) K =Tn2N

n(I1 [ I2 [ [ Ir).

Remark 2.2. If k is not an integer, say k = m+ ˛, with m 1 integer and 0 < ˛ < 1

we assume that is Cm and (m) is ˛-Hölder.

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 737

We say that fI1; I2; : : : ; Irg is aMarkov partition for K and that K is defined by .

Remark 2.3. In general, we say that a set X R is a Cantor set if X is compact,without isolated points and with empty interior. Cantor sets in R are homeomorphic tothe classical ternary Cantor set K1/3 of the elements of [0; 1] which can be written inbase 3 using only digits 0 and 2. The set K1/3 is itself a regular Cantor set, defined bythe map : [0; 1/3] [ [2/3; 1] ! R given by [x] = 3x b3xc.

An interval of the construction of the regular Cantor setK is a connected componentof n(Ij ) for some n 2 N, j r .

Given s 2 [1; k] and another regular Cantor set K, we say that K is close toK in theC s topology if K has a Markov partition fI1; I2; : : : ; Irg such that the interval Ij hasendpoints close to the endpoints of Ij , for 1 j r and K is defined by a C s map which is close to in the C s topology.

The C 1+-topology is such that a sequence n converges to if there is some ˛ > 0

such that n is C 1+˛ for every n 1 and n converges to in the C 1+˛-topology.The concept of stable intersection of two regular Cantor sets was introduced in Mor-

eira [1996]: two Cantor sets K1 and K2 have stable intersection if there is a neighbour-hood V of (K1; K2) in the set of pairs of C 1+-regular Cantor sets such that (eK1; eK2) 2

V ) eK1 \ eK2 ¤ ¿.In the same paper conditions based on renormalizations were introduced to ensure

stable intersections, and applications of stable intersections to homoclinic bifurcationswere obtained: roughly speaking, if some translations of the stable and unstable regularCantor sets associated to the horseshoe at the initial bifurcation parameter = 0 havestable intersection then the set f > 0 j ' presents persistent homoclinic tangenciesghas positive Lebesgue density at = 0. It was also shown that this last phenomenoncan coexist with positive density of hyperbolicity in a persistent way.

Besides, the following question was posed in Moreira [ibid.]: Does there exist adense (and automatically open) subset U of

Ω1 = f(K1; K2);K1; K2 areC1regular Cantor sets andHD(K1)+HD(K2) > 1g

such that (K1; K2) 2 U ) 9 t 2 R such that (K1; K2 + t) has stable intersection? Apositive answer to this question implies a strong version of Palis’ conjecture. Indeed,K1 K2 = ft 2 R j K1 \ (K2 + t) ¤ ¿g, so, if (K1; K2 + t) has stable intersectionthen t belongs persistently to the interior of K1 K2.

Moreira and Yoccoz [2001] gave an affirmative answer to this question, proving thefollowing

Theorem 2.4. There is an open and dense set U Ω1 such that if (K1; K2) 2 U,then Is(K1; K2) is dense in K1 K2 andHD((K1 K2)nIs(K1; K2)) < 1, where

Is(K1; K2) := ft 2 R j K1 and (K2 + t) have stable intersectiong:

738 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

Jean-Christophe Yoccoz

The same result works if we replace stable intersection by d -stable intersection,which is defined by asking that any pair (eK1; eK2) in some neighbourhood of (K1; K2)

satisfiesHD(eK1 \ eK2) d : most pairs of Cantor sets (K1; K2) 2 Ω1 have d -stableintersection for any d < HD(K1) +HD(K2) 1.

The open set U mentioned in the above theorem is very large inΩ1 in the sense thatgeneric n-parameter families in Ω1 are actually contained in U.

The proof of this theorem depends on a sufficient condition for the existence of stableintersections of two Cantor sets, related to a notion of renormalization, based on the factthat small parts of regular Cantor sets are diffeomorphic to the whole set: the existenceof a recurrent compact set of relative positions of limit geometries of them. Roughlyspeaking, it is a compact set of relative positions of regular Cantor sets such that, forany relative position in such a set, there is a pair of (small) intervals of the constructionof the Cantor sets such that the renormalizations of the Cantor sets associated to theseintervals belong to the interior of the same compact set of relative positions.

The main result is reduced to prove the existence of recurrent compact sets of relativepositions for most pairs of regular Cantor sets whose sum of Hausdorff dimensions islarger than one. A central argument in the proof of this fact is a probabilistic argumentà la Erdős: we construct a family of perturbations with a large number of parameters

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 739

and show the existence of such a compact recurrent set with large probability in theparameter space (without exhibiting a specific perturbation which works).

Another important ingredient in the proof is the Scale Recurrence Lemma, which,under mild conditions on the Cantor sets (namely that at least one of them is not essen-tially affine), there is a recurrent compact set for renormalizations at the level of relativescales of limit geometries of the Cantor sets. This lemma is the fundamental tool in thepaper Moreira [2016], in which it is proved that, under the same hypothesis above, ifKandK 0 are C 2-regular Cantor sets, thenHD(K +K 0) = minf1;HD(K)+HD(K 0)g.

An important result in fractal geometry which is used in Moreira and Yoccoz [2001]is the famous Marstrand’s theorem (Marstrand [1954]), according to which, given aBorel set X R2 withHD(X) > 1 then, for almost every 2 R, (X) has positiveLebesgue measure, where : R2 ! R is given by (x; y) = x y. In particular,if K1 and K2 are regular Cantor sets with HD(K1) +HD(K2) > 1 then, for almostevery 2 R, K1 K2 has positive Lebesgue measure. Moreira and Yuri Lima gavecombinatorial alternative proofs of Marstrand’s theorem, first in the case of Cartesianproducts regular Cantor sets (Y. Lima and Moreira [2011a]) and then in the general case(Y. Lima and Moreira [2011b]).

In Moreira and Yoccoz [2010], Moreira and Yoccoz proved the following fact con-cerning generic homoclinic bifurcations associated to two dimensional saddle-type hy-perbolic sets (horseshoes) with Hausdorff dimension bigger than one: typically thereare translations of the stable and unstable Cantor sets having stable intersection, and soit yields open sets of stable tangencies in the parameter line with positive density at theinitial bifurcation value. Moreover, the union of such a set with the hyperbolicity setin the parameter line generically has full density at the initial bifurcation value. Thisextends the results of Palis Jr. and Yoccoz [1994].

The situation is quite different in the C 1-topology, in which stable intersections donot exist:

Theorem 2.5 (Moreira [2011]). Given any pair (K;K 0) of regular Cantor sets, we canfind, arbitrarily close to it in the C 1 topology, pairs (K; K 0) of regular Cantor sets withK \ K 0 = ¿.

Moreover, for generic pairs (K;K 0) of C 1-regular Cantor sets, the arithmetic differ-ence K K 0 has empty interior (and so is a Cantor set).

The main technical difference between the C 1 case and the C 2 (or even C 1+˛) casesis the lack of bounded distortion of the iterates of in the C 1 case, and this fact isfundamental for the proof of the previous result.

The previous result may be used to show that there are no C 1 robust tangencies be-tween leaves of the stable and unstable foliations of respectively two given hyperbolichorseshoes Λ1;Λ2 of a diffeomorphism of a surface. This is also very different fromthe situation in the C1 topology - for instance, in Moreira and Yoccoz [2010] it isproved that, in the unfolding of a homoclinic or heteroclinic tangency associated to twohorseshoes, when the sum of the correspondent stable and unstable Hausdorff dimen-sions is larger than one, there are generically stable tangencies associated to these twohorseshoes. This result is done in the following

740 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

Theorem 2.6 (Moreira [2011]). Given a C 1 diffeomorphism of a surfaceM havingtwo (non necessarily disjoint) horseshoes Λ1;Λ2, we can find, arbitrarily close to it inthe C 1 topology, a diffeomorphism of the surface for which the horseshoes Λ1;Λ2

have hyperbolic continuations Λ1; Λ2, and there are no tangencies between leaves of thestable and unstable foliations of Λ1 and Λ2, respectively. Moreover, there is a genericset R of C 1 diffeomorphisms ofM such that, for every 2 R, there are no tangenciesbetween leaves of the stable and unstable foliations ofΛ1;Λ2, for any horseshoesΛ1;Λ2

of .

Since stable intersections of Cantor sets are the main known obstructions to densityof hyperbolicity for diffeomorphisms of surfaces, the previous result gives some hopeof proving density of hyperbolicity in the C 1 topology for diffeomorphisms of surfaces,a well-known question by Smale. In particular in the work Matheus, Moreira, and Pu-jals [2013] on a family of two-dimensional maps (the so-called Benedicks–Carleson toymodel for Hénon dynamics) in which we prove that in this family there are diffeomor-phisms which present stable homoclinic tangencies (Newhouse’s phenomenon) in theC 2-topology but their elements can be arbitrarily well approximated in theC 1-topologyby hyperbolic maps.

2.3 Palis–Yoccoz theorem on non-uniformly hyperbolic horseshoes. In a majorwork by Palis Jr. and Yoccoz [2009]; (see also Palis Jr. and Yoccoz [2001b] and Palis Jr.and Yoccoz [2001a]), they propose to advance considerably the current knowledge onthe topic of bifurcations of heteroclinic cycles for smooth (C1) parametrized familiesfgt ; t 2 Rg of surface diffeomorphisms. They assume that gt is hyperbolic for t < 0

and jt j small and that a quadratic tangency q is formed at t = 0 between the stable andunstable lines of two periodic points, not belonging to the same orbit, of a (uniformlyhyperbolic) horseshoeΛ and that such lines cross each other with positive relative speedas the parameter evolves, starting at t = 0 near the point q. They also assume that, insome neighbourhoodW of the union ofΛwith the orbit of tangencyO(q), the maximalinvariant set for g0 = gt=0 is Λ [O(q), where O(q) denotes the orbit of q for g0.

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 741

A main novelty is that they allowed the Hausdorff dimension HD(Λ) to be biggerthan one: they assume HD(Λ) to be bigger than one, but not much bigger (more pre-cisely, they assume that, if ds and du are the Hausdorff dimensions of, respectively, thestable and unstable Cantor sets of g0 then (ds + du)

2 + max(ds; du)2 < ds + du +

max(ds; du)). Then, for most small values of t , gt is a “non-uniformly hyperbolic”horseshoe inW , and so gt has no attractors nor repellors inW . Most small values of t ,and thus most gt , here means that t is taken in a set of parameter values with Lebesguedensity one at t = 0.

The construction of non-uniformly hyperbolic horseshoes for most parameters is ahighly non-trivial counterpart of Yoccoz’ proof Yoccoz [2015] (based on the so-calledYoccoz puzzles) of the celebrated Jakobson’s theorem in the context of heteroclinic ex-plosions.

Of course, Palis and Yoccoz do not consider their result as the end of the line. Indeed,they expected the same results to be true for all cases 0 < HD(Λ) < 2. However, toachieve that, it seems that their methods need to be considerably sharpened: it would benecessary to study deeper the dynamical recurrence of points near tangencies of higherorder (cubic, quartic...) between stable and unstable curves. They also expected theirresults to be true in higher dimensions.

Finally, they hoped that the ideas introduced in that work might be useful in broadercontexts. In the horizon lies a famous question concerning the standard family of areapreserving maps (which is the family (fk)k2R of diffeomorphisms of the torus T 2 =

R2/Z2 given by fk(x; y) = (y + 2x + k sin(2x); x) (mod 1)): can we find setsof positive Lebesgue measure in the parameter space such that the corresponding mapsdisplay non-zero Lyapunov exponents in sets of positive Lebesgue probability in phasespace?

In a recent work, Matheus, Moreira and Palis applied the results of Palis Jr. andYoccoz [2009] to prove the following

Theorem 2.7. There exists k0 > 0 such that, for all jkj > k0 and " > 0, the subset ofparameters r 2 R such that jr kj < " and fr exhibits a non-uniformly hyperbolichorseshoe (in the sense of Palis Jr. and Yoccoz [ibid.]) has positive Lebesgue measure.

2.4 Hyperbolic sets in higher dimensions. The fractal geometric properties of horse-shoes of surface diffeomorphisms are quite regular and well-understood. As we men-tioned before, a horseshoe Λ in ambient dimension 2 is locally diffeomorphic to a carte-sian product Ks Ku, where Ks and Ku are regular Cantor sets on the real line. Mc-Cluskey andManning proved thatHD(Λ) is continuous in theC 1topology (later Palisand Viana gave a more geometrical proof of this fact).

In ambient dimension larger than two, much less is known about the geometry ofhorseshoes. In general, if Λ is such a horseshoe, the Hausdorff dimension HD(Λ) isnot always continuous, as shown by Bonatti, Díaz and Viana. A natural question iswhetherHD(Λ) is “typically” continuous. Another natural question is what can be saidabout the fractal geometry of stable and unstable Cantor sets of horseshoes in arbitrarydimensions. These Cantor sets are no more conformal sets with bounded geometry,which was the case of stable and unstable Cantor sets of horseshoes in surfaces. In what

742 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

follows, we will discuss some recent results on the geometry of horseshoes in arbitrarydimensions.

In the first work, in collaboration with J. Palis and M. Viana, we generalize an im-portant lemma from Moreira, Palis Jr., and Viana [2001]: given a horseshoe Λ whosestable spaces have dimension ` we define a family of fractal dimensions (the so-calledupper stable dimensions) d

(j )

s (Λ); 1 j ` which satisfy d(1)

s (Λ) d(2)

s (Λ)

d(`)

s (Λ) HD(Λ \ W s(x));8x 2 Λ. We define ds(Λ) = d(`)s (Λ). When

r < d(r)

s (Λ) r + 1 we have ds(Λ) = d(r)

s (Λ) These definitions are inspired inthe affinity dimensions, introduced by Falconer. We have analogous definitions for up-per unstable dimensions. We prove the following results about these dimensions: given1 r ` and " > 0 there is a "smallC1 perturbation of the original diffeomorphismfor which the hyperbolic continuation ofΛ has a subhorseshoe Λwhich has strong-stablefoliations of codimensions j for 1 j r and which satisfies d

(r)

s (Λ) > d(r)

s (Λ) ".In the second work in progress, in collaboration with W. Silva (which extends a

previous joint work in codimension 1 - see Moreira and Silva [2012]), we prove that if ahorseshoe Λ has strong stable foliations of codimensions j for 1 j r and satisfiesds(Λ) > r (which is equivalent to d

(r)

s (Λ) > r) then it has a small C1 perturbationwhich contains a blender of codimension r : in particular C 1 images of stable Cantorsets of it (of the type Λ \ W s(x)) in Rr will typically have persistently non-emptyinterior. We also expect to prove that the Hausdorff dimension of these stable Cantorsets typically coincide with ds(Λ), and this dimension depends continuously on Λ onthese assumptions, which would imply typical continuity of Hausdorff dimensions ofstable and unstable Cantor sets of horseshoes.

Let us recall two main ingredients in the proof by Moreira and Yoccoz of Palis’ con-jecture:

• A recurrent compact set criterion for stable intersections (which implies that arith-metic differences persistently contain intervals).

• An application of Erdős probabilistic method: a family ofC1 small perturbationsof a regular Cantor set (the second Cantor set is fixed) with a large number ofparameters such that for most parameters there is a recurrent compact set for thecorresponding pair of Cantor sets.

A variation of these ingredients is present in the proof of these results in collaborationof W. Silva: we develop a recurrent compact set criterion which implies that a givenhorseshoe (with a strong-stable foliation of codimension r) is a r-codimensional blender,and we prove that, if a horseshoe has a strong-stable foliation of codimension r andsatisfies ds(Λ) > r (which is equivalent to d

(r)

s (Λ) > r), then there is a small C1

perturbation of it which has a recurrent compact set. In order to do this, we use theprobabilistic method: we construct a family of perturbations with a large number ofparameters, and show that for most parameters there is such a recurrent compact set.

An important geometrical tool in the proof of these results is the following general-ization ofMarstrand’s theorem, which was proved in works by López, Moreira, Romañaand Silva:

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 743

Theorem 2.8. Let X be a compact metric space, (Λ;P ) a probability space and :

Λ X ! Rr a measurable function. Informally, one can think of () = (; )

as a family of projections parameterized by . We assume that for some positives realnumbers ˛ and C the following transversality property is satisfied:

(1) P [ 2 Λ : d ((x1); (x2)) ıd (x1; x2)˛] Cır

for all ı > 0 and all x1; x2 2 X . Assume that dimX > ˛r . Then Leb((X)) > 0 fora.e. 2 Λ and

RΛLeb((X))1dP < +1.

3 Fractal Geometry and Diophantine Approximations

3.1 The classical Markov and Lagrange spectra. The results discussed in the pre-vious section on regular Cantor sets have, somewhat surprisingly, deep consequencesin number theory, which we discuss below. We begin by introducing the classical La-grange spectrum.

Let ˛ be an irrational number. According to Dirichlet’s theorem, the inequalityj˛

pq

j < 1q2

has infinitely many rational solutions pq. Hurwitz improved this result

by proving that j˛ pq

j < 1p5q2

also has infinitely many rational solutions pqfor any

irrational ˛, and thatp5 is the largest constant that works for any irrational ˛. However,

for particular values of ˛ we can improve this constant.More precisely, we define k(˛) := supfk > 0 j j˛

pq

j < 1kq2

has infinitelymany rational solutions p

qg = lim supp;q!+1 (qjq˛ pj)1. We have k(˛)

p5,

8˛ 2 R n Q and k1+

p5

2

=

p5. We will consider the set L = fk(˛) j ˛ 2 R n Q,

k(˛) < +1g.This set is called the Lagrange spectrum. Hurwitz’s theorem determines the smallest

element of L, which isp5. This set L encodes many diophantine properties of real

numbers. It is a classical subject the study of the geometric structure of L. Markovproved in 1879 (Markoff [1880]) that

L \ (1; 3) = fk1 =p5 < k2 = 2

p2 < k3 =

p221

5< : : : g

where kn is a sequence (of irrational numbers whose squares are rational) convergingto 3.

The elements of the Lagrange spectrum which are smaller than 3 are exactly thenumbers of the form

q9

4z2

where z is a positive integer for which there are otherpositive integers x; y such that 1 x y z and (x; y; z) is a solution of theMarkovequation

x2 + y2 + z2 = 3xyz:

Since this is a quadratic equation in x (resp. in y), whose sum of roots is 3yz (resp.3xz), given a solution (x; y; z) with x y z, we also have the two other solutions(y; z; 3yz x) and (x; z; 3xz y) - it is possible to prove that all solutions of theMarkov equation appear in the following tree:

744 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

(1; 1; 1)

(1; 1; 2)

(1; 2; 5)

(1; 5; 13)

(1; 13; 34) (5; 13; 194)

(2; 5; 29)

(2; 29; 169) (5; 29; 433)

An important open problem related to Markov’s equation is the Unicity Problem,formulated by Frobenius about 100 years ago: for any positive integers x1; x2; y1; y2; zwith x1 y1 z and x2 y2 z such that (x1; y1; z) and (x2; y2; z) are solutionsof Markov’s equation we always have (x1; y1) = (x2; y2)?

If the Unicity Problem has an affirmative answer then, for every real t < 3, its pre-image k1(t) by the function k above consists of a single GL2(Z)-equivalence class(this equivalence relation is such that

˛ ∼a˛ + b

c˛ + d;8a; b; c; d 2 Z; jad bcj = 1:)

Despite the “beginning” of the set L is discrete, this is not true for the whole setL. As we mentioned in the introduction, M. Hall proved in 1947 (Hall [1947]) that ifC (4) is the regular Cantor set formed by the numbers in [0; 1] whose coefficients in thecontinued fractions expansion are bounded by 4, then one has C (4) + C (4) = [

p2

1; 4(p21)]. This implies thatL contains a whole half line (for instance [6;+1)), and

G. Freĭman determined in 1975 (Freıman [1975]) the biggest half line that is containedin L, which is [cF ;+1), with

cF =2221564096 + 283748

p462

491993569Š 4; 52782956616 : : : :

These last two results are based on the study of sums of regular Cantor sets, whoserelationship with the Lagrange spectrum will be explained below.

Sets of real numbers whose continued fraction representation has bounded coeffi-cients with some combinatorial constraints, asC (4), are often regular Cantor sets, whichwe call Gauss–Cantor sets (since they are defined by restrictions of the Gauss mapg(x) = f1/xg from (0; 1) to [0; 1) to some convenient union of intervals).

We represent below the graphics of the Gauss map g(x) = f1x

g.If the continued fraction of ˛ is

˛ = [a0; a1; a2; : : : ]def= a0 +

1

a1 +1

a2 + :::

:

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 745

y = g(x) =˚1x

then we have the following formula for k(˛):

k(˛) = lim supn!1

(˛n + ˇn);

where ˛n = [an; an+1; an+2; : : : ] and ˇn = [0; an1; an2; : : : ; a1]:

The previous formula follows from the equality

j˛ pn

qnj =

1

(˛n+1 + ˇn+1)q2n; 8n 2 N;

where

pn/qn = [a0; a1; a2; : : : ; an] = a0 +1

a1 +1

a2 + :::+ 1an

; n 2 N

are the convergents of the continued fraction of ˛.There are many results which relate the dynamics of the Gauss map with the be-

haviour of continued fractions. For instance, the Khintchine–Lévy theorem, which fol-lows from techniques of Ergodic Theory, states that, for (Lebesgue) almost every ˛ 2 R,

limn!1

npqn = e

2/12 log2:

Remark: The following elementary general facts on Diophantine approximationsof real numbers show that the best rational approximations of a given real number aregiven by convergents of its continued fraction representation: For every n 2 N, ˇ

˛ pn

qn

ˇ<

1

2q2nor

ˇ˛

pn+1

qn+1

ˇ<

1

2q2n+1

(moreover, for every positive integer n, there is k 2 fn 1; n; n+ 1g with j˛ pk

qkj <

1p5q2n

). If

ˇ˛

pq

ˇ< 1

2q2then p

qis a convergent of the continued fraction of ˛.

746 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

This formula for k(˛) implies that we have the following alternative definition ofthe Lagrange spectrum L:

Let Σ = (N)Z be the set of all bi-infinite sequences of positive integers. If =

(an)n2Z 2 Σ, let ˛n = [an; an+1; an+2; : : : ] and ˇn = [0; an1; an2; : : : ];8n 2 Z.We define f () = ˛0 + ˇ0 = [a0; a1; a2; : : : ] + [0; a1; a2; : : : ]. We have

L = flim supn!1

f (n); 2 Σg;

where : Σ ! Σ is the shift defined by ((an)n2Z) = (an+1)n2Z.Let us define the Markov spectrumM by

M = fsupn2Z

f (n); 2 Σg:

It also has an arithmetical interpretation, namely

M = f( inf(x;y)2Z2n(0;0)

jf (x; y)j)1; f (x; y) = ax2 + bxy + cy2; b2 4ac = 1g:

It is well-known (see Cusick and Flahive [1989]) that M and L are closed sets of thereal line and L M .

We have the following result about the Markov and Lagrange spectra:

Theorem 3.1. (Moreira [2018]; see also Matheus [2017]) Given t 2 R we have

HD(L \ (1; t)) = HD(M \ (1; t)) =: d (t)

and d (t) is a continuous surjective function from R to [0; 1]. Moreover:i) d (t) = minf1; 2D(t)g, where D(t) := HD(k1(1; t)) = HD(k1(1; t ])

is a continuous function from R to [0; 1).ii) maxft 2 R j d (t) = 0g = 3

iii) d (p12) = 1.

A fundamental tool in the proof of this result is the theorem below.We say that a C 2-regular Cantor set on the real line is essentially affine if there is a

C 2 change of coordinates for which the dynamics that defines the corresponding Cantorset has zero second derivative on all points of that Cantor set. TypicalC 2-regular Cantorsets are not essentially affine.

The scale recurrence lemma, which is the main technical lemma of Moreira andYoccoz [2001], can be used in order to prove the following

Theorem 3.2. (Moreira [2016]) If K and K 0 are regular Cantor sets of class C 2 andK is non essentially affine, thenHD(K +K 0) = minfHD(K) +HD(K 0); 1g:

Remark 3.3. There is a presentation of a version of this result (with a slightly differenthypothesis) in Moreira [1999]. That version is also proved by Hochman and Shmerkin[2012].

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 747

The results of Markov, Hall and Freĭman mentioned above imply that the Lagrangeand Markov spectra coincide below 3 and above cF . Nevertheless, Freıman [1968]showed in 1968 that M n L ¤ ¿ by exhibiting a number ' 3:1181 2 M n L.Moreira andMatheus described the structure of the intersection ofM nLwith an intervalnear this point, and proved that it has positive Hausdorff dimension (see Matheus andMoreira [2019]).

In 1973, Freıman [1973] showed that

˛1 := 0(A1) := [2; 12; 23; 1; 2] + [0; 1; 23; 12; 2; 1; 2] 2 M n L

In a similar vein, Theorem 4 in Chapter 3 of Cusick and Flahive [1989] book assertsthat

˛n := 0(An) := [2; 12; 23; 1; 2] + [0; 1; 23; 12; 2; 1; 2n; 1; 2; 12; 23] 2 M n L

for all n 4. In particular, ˛1 is not isolated inM n L.In collaboration with C. Matheus, we proved the following results aboutM n L:Let X be the Cantor set

(2) X := f[0; ] : 2 f1; 2gN not containing the subwords in P g

where

P := f21212; 21213; 13212; 121212; 122121; 231212221; 122122123; 123121222; 221221231g

Also, let

b1 := [2; 12; 23; 1; 2] + [0; 1; 23; 12; 2] =

p18229

41= 3:2930442439 : : :

and

B1 := [2; 1; 1; 23; 1; 2; 12; 2; 12; 2] + [0; 1; 23; 12; 2; 1; 23; 12; 2; 1; 22; 1; 23; 1; 2; 12; 2; 12; 2]

= 3:2930444814 : : :

Theorem 3.4. (Matheus and Moreira [2017b]) fb1; B1g L, L \ (b1; B1) = ¿andHD((M n L) \ (b1; B1)) = HD(X).

We implemented the algorithm of Jenkinson and Pollicott [2018] and we obtainedthe heuristic approximation HD(X) = 0:4816 . We also we exhibited a Cantor setK(f1; 22g) X whose Hausdorff dimension can be easily (and rigorously) estimatedas 0:353 < HD(K(f1; 22g)) < 0:35792 via some classical arguments explained inPalis Jr. and Takens [1987] book.

By exploiting the arguments establishing the above Theorem, we are able to exhibitnew numbers inM n L, including a constant c 2 M n L with c > ˛4:

Proposition 3.5. The largest element of (M n L) \ (b1; B1) is

c =77 +

p18229

82+

17633692 p151905

24923467= 3:29304447990138 : : : :

748 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

We also found elements ofM nL larger thanp12, disproving a conjecture by Cusick,

and giving a new lower estimate for the Hausdorff dimension ofM n L (see Matheusand Moreira [2018]):

Proposition 3.6. The Hausdorff dimension of (M n L) \ (3:7; 3:71) is larger than0:53128. This intersection includes the element

(3) [3; 2; 2; 2; 1; 2; 3; 3; 2; 2; 2; 1; 2; 2; 1; 2; 1; 2; 1; 1; 2] + [0; 3; 2; 1; 2; 2; 2; 3; 3] =

=7940451225305

p3

2326589591051+

483 +p330629

310= 3:70969985975 : : : :

To our best knowledge, this is the largest known element ofM n L.

We also proved (Matheus and Moreira [2017a]) thatM n L doesn’t have full Haus-dorff dimension:

Theorem 3.7. HD(M n L) < 0:986927.

One can get better heuristic bounds for HD(M n L) thanks to the several methodsin the literature to numerically approximate the Hausdorff dimension of Cantor sets ofnumbers with prescribed restrictions of their continued fraction expansions. By imple-menting the “thermodynamical method” introduced in Jenkinson and Pollicott [2001],we obtained the heuristic boundHD(M n L) < 0:888.

Our proof of Theorem 3.7 relies on the control of several portions ofM nL in termsof the sum-set of a Cantor set associated to continued fraction expansions prescribed bya “symmetric block” and a Cantor set of irrational numbers whose continued fractionexpansions live in the “gaps” of a “symmetric block”. As it turns out, such a controlis possible thanks to our key technical Lemma saying that a sufficiently large Markovvalue given by the sum of two continued fraction expansions systematically meeting a“symmetric block” must belong to the Lagrange spectrum.

It follows thatM n L has empty interior, and so, sinceM and L are closed subsetsof R, int(M ) = int(L) L M . In particular, we have the following

Corollary 3.8. int(M ) = int(L).

As a consequence, we recover the fact, proved in Freıman [1975], that the biggesthalf-line contained inM coincides with the biggest half-line [cF ;1) contained in L.

3.2 Other results on the fractal geometry of Diophantine approximations. In col-laboration with Y. Bugeaud (see Bugeaud and Moreira [2011]), we proved some resultson sets of exact approximation order by rational numbers:

For a function Ψ : (0;+1) ! (0;+1), let

K(Ψ) :=

2 R :

ˇ

p

q

ˇ< Ψ(q) for infinitely many rational numbers

p

q

denote the set of Ψ-approximable real numbers and let

Exact(Ψ) := K(Ψ) n[m2

K(1 1/m)Ψ

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 749

be the set of real numbers approximable to order Ψ and to no better order.The lower order at infinity (g) of a function g : (0;+1) ! (0;+1) is defined

by

(g) = lim infx!+1

logg(x)log x

:

We say that a function Ψ : (0;+1) ! (0;+1) satisfies assumption () if Ψ(x) =

o(x2) and there exist real numbers c, c and n0 with 1 c < 4 such that, if the positiveintegers m, n satisfy m > n n0, then Ψ(m)mc cΨ(n)nc . We emphasize that thereal number c occurring in () may be negative.

Theorem 3.9. Let Ψ : (0;+1) ! (0;+1) be a function satisfying assumption ().Then the set Exact(Ψ) is uncountable.

Theorem 3.10. Let Ψ : (0;+1) ! (0;+1) be a function satisfying assumption ().If denotes the lower order at infinity of the function 1/Ψ, then

dimExact(Ψ) = dimK(Ψ) =2

:

In another work collaboration with Y. Bugeaud (see Bugeaud and Moreira [2016]),we proved that there are no typical real numbers from the point of view of Diophan-tine approximations, in a sense that we describe in what follows. Let Ψ be an ap-plication from the set of positive integers into the set of nonnegative real numbers.Khintchine established that, if the function q 7! q2Ψ(q) is non-increasing and theseries

Pq1 qΨ(q) diverges, then the set K(Ψ) has full Lebesgue measure (Beres-

nevich, Dickinson and Velani proved later the same result assuming that Ψ is just non-increasing). We show that, for almost every real number ˛, there is a function Ψ whichsatisfies good “regularity” conditions (on the speed of decreasing of Ψ) - for instanceq 7! q2Ψ(q) is non-increasing, such that the series

Pq1 qΨ(q) diverges but the in-

equality j˛ p/qj < Ψ(q) has no rational solution p/q.Khintchine also showed that if the series

Pq1 qΨ(q) converges, then the set K(Ψ)

has zero Lebesgue measure. We show that, for almost every real number ˛, there isa function Ψ which satisfies good “regularity” conditions (for instance q 7! q2Ψ(q)

is non-increasing), such that the seriesPq1 qΨ(q) converges but the inequality j˛

p/qj < Ψ(q) has infinitely many rational solutions p/q.We also compute Hausdorff dimensions of sets of exceptions to our results (in terms

of the regularity conditions on Ψ):

Theorem 3.11. Let be a real number in the interval [1; 2]. We define X the set ofirrational numbers such that, for every function Ψ with q 7! qΨ(q) non-increasingand satisfying

1Xq=1

qΨ(q) = +1;

the inequality ˇ

p

q

ˇ< Ψ(q)

750 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

has infinitely many rational solutions. We proved that X1 is empty and that, for 2

(1; 2], the Hausdorff dimension of X is /2 (and so, when varies, assume all valuesin the interval (1/2; 1]).

Theorem 3.12. Let b be a positive real number. We define Yb as the set of the irra-tional numbers such that, for every function Ψ withq 7! q(log log q)b log logqΨ(q)

non-increasing and satisfying1Xq=1

qΨ(q) = +1;

the inequality ˇ

p

q

ˇ< Ψ(q)

has infinitely many rational solutions. Then the Hausdorff dimension of Yb is 11+e1/b

(and so, when b varies, assume all values in the interval (0; 1/2)).

One of the tools we used in the proof of this last theorem is the results from Łuczak[1997]: given b; c > 1 define

Ξ(b; c) = f = [0; a1; a2; :::] 2 [0; 1] : an cbn for every n 1g

and

Ξ(b; c) = f = [0; a1; a2; :::] 2 [0; 1] : an cbn for infinitely many n 1g

Then HD(Ξ(b; c)) = HD(Ξ(b; c)) = 11+b

. In one direction, a more precise resultis proved in Feng, Wu, Liang, and Tseng [1997], according to which the Hausdorffdimension of the set

K := f = [0; a1; a2; :::] 2 [0; 1] : exp(bn) an 3 exp(bn); for every n 1g

is equal to 1

1+b. We notice that an older result fromGood [1941] states that theHausdorff

dimension of the set f = [0; a1; a2; :::] 2 [0; 1] : lim an = 1g is 1/2.

4 Back to Dynamical Systems (and Differential Geometry)

As we have seen, the sets M and L can be defined in terms of symbolic dynamics.Inspired by these characterizations, we may associate to a dynamical system togetherwith a real function generalizations of the Markov and Lagrange spectra as follows:

Definition 4.1. Given a map : X ! X and a function f : X ! R, we define theassociated dynamical Markov and Lagrange spectra asM (f; ) = fsupn2Nf (

n(x)); x 2 Xg andL(f; ) = flimsupn!1f (

n(x)); x 2 Xg, respectively.Given a flow ('t )t2R in a manifold X , we define the associated dynamical Markov

and Lagrange spectra asM (f; ('t )) = fsupt2Rf ('t (x)); x 2 Xg and

L(f; ('t )) = flimsupt!1f ('t (x)); x 2 Xg, respectively.

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 751

In awork in collaborationwithA. Cerqueira andC.Matheus (see Cerqueira, Matheus,and Moreira [2018]), we prove the following result, which generalizes a correspondingfact in the context of the classical Markov and Lagrange spectra:

Lemma 4.2. Let ('; f ) be a generic pair, where ' : M 2 ! M 2 is a diffeomorphismwith Λ M 2 a hyperbolic set for ' and f : M ! R is C 2. Let s; u be theprojections of the horseshoe Λ to the stable and unstable regular Cantor sets Ks; Kuassociated to it (along the unstable and stable foliations of Λ). Given t 2 R, we define

Λt =\m2Z

'm(fp 2 Λjf (p) tg);

Kst = s(Λt ); Kut = u(Λt ):

Then the functions ds(t) = HD(Kst ) and du(t) = HD(Kut ) are continuous and coin-cide with the corresponding box dimensions.

The following result is a consequence of the scale recurrence lemma of Moreira andYoccoz [2001] (see Moreira [2016]):

Lemma 4.3. Let ('; f ) be a generic pair, where ' : M 2 ! M 2 is a diffeomorphismwith Λ M 2 a hyperbolic set for ' and f :M ! R is C 2. Then

HD(f (Λ)) = min(HD(Λ); 1):

Moreover, ifHD(Λ) > 1 then f (Λ) has persistently non-empty interior.

Using the previous lemmas we prove a generalization of the results on dimensionsof the dynamical spectra:

Theorem 4.4. (Cerqueira, Matheus, and Moreira [2018]) Let ('; f ) be a generic pair,where ' : M 2 ! M 2 is a conservative diffeomorphism with Λ M 2 a hyperbolic setfor ' and f :M ! R is C 2. Then

HD(L(f;Λ) \ (1; t)) = HD(M (f;Λ) \ (1; t)) =: d (t)

is a continuous real function whose image is [0;min(HD(Λ); 1)].

(here we use the notation L(f;Λ) := L(f; 'jΛ)).In Hubert, Marchese, and Ulcigrai [2015], P. Hubert, L. Marchese and C. Ulcigrai

introduced in a similar way to the above generalizations the Lagrange spectra of closed-invariant loci for the action of SL(2;R) on the moduli space of translation surfaces, inthe context of Teichmüller dynamics, and proved that several of these spectra contain aHall’s ray.

Moreira and Romaña proved the following result on Markov and Lagrange spectrafor horseshoes:

Theorem 4.5 (Romaña Ibarra and Moreira [2017a]). Let Λ be a horseshoe associatedto a C 2-diffeomorphism ' such that HD(Λ) > 1. Then there is, arbitrarily close to' a diffeomorphism '0 and a C 2-neighborhood W of '0 such that, if Λ denotes the

752 CARLOS GUSTAVO TAMM DE ARAUJO MOREIRA

continuation ofΛ associated to 2 W , there is an open and dense setH C 1(M;R)

such that for all f 2 H , we have

int L(f;Λ ) ¤ ¿ and intM (f;Λ ) ¤ ¿;

where intA denotes the interior of A.

Recently, D. Lima and Moreira [2018] proved that, for typical pairs (f;Λ) as in theabove theorem for which ' is conservative,

supft 2 RjHD(M (f;Λ)\(1; t)) < 1g = infft 2 Rjint(L(f;Λ)\(1; t)) ¤ ¿g;

and Moreira [2017] proved that, for typical pairs (f;Λ) as above, the minima of thecorresponding Lagrange and Markov dynamical spectra coincide and are given by theimage of a periodic point of the dynamics by the real function, solving a question byYoccoz.

The classical Markov and Lagrange spectra can also be characterized as sets of max-imum heights and asymptotic maximum heights, respectively, of geodesics in the mod-ular surface N = H2/PSL(2;Z). Moreira and Romaña extend in Romaña Ibarra andMoreira [2015] the fact that these spectra have non-empty interior to the context ofnegative, non necessarily constant curvature as follows:

Theorem 4.6. (Romaña Ibarra andMoreira [ibid.]) LetM provided with ametric g0 bea complete noncompact surfaceM with finite Gaussian volume and Gaussian curvaturebounded between two negative constants, i.e., if KM denotes the Gaussian curvature,then there are constants a; b > 0 such that

a2 KM b2 < 0:

Denote by SM its unitary tangent bundle and by its geodesic flow.Then there is ametric g close to g0 and a dense andC 2-open subsetH C 2(SM;R)

such thatintM (f; g) ¤ ¿ and int L(f; g) ¤ ¿

for any f 2 H, where g is the vector field defining the geodesic flow of the metric g.Moreover, if X is a vector field sufficiently close to g then

intM (f;X) ¤ ¿ and int L(f;X) ¤ ¿

for any f 2 H.

Recently, in collaboration with Pacífico and Romaña (see Moreira, Pacifico, andRomaña Ibarra [n.d.]), we proved an analogous result for Lorenz flows. Romaña alsoproved (Romaña Ibarra and Moreira [2017b]) a corresponding result for Anosov flowsin compact 3-dimensional manifolds.

Combining the techniques of this result with those of the above results in collabo-ration with A. Cerqueira and C. Matheus, Cerqueira, Moreira and Romaña proved thefollowing:

DYNAMICAL SYSTEMS, FRACTALS AND APPROXIMATIONS 753

Theorem4.7. (Cerqueira, Moreira, and Romaña Ibarra [2017]) Let ('; f ) be a genericpair, where ' : M 2 ! M 2 is a conservative diffeomorphism with Λ M 2 a hyper-bolic set for ' and f :M ! R is C 2. Then

HD(L(f;Λ) \ (1; t)) = HD(M (f;Λ) \ (1; t)) =: d (t)

is a continuous real function whose image is [0;min(HD(Λ); 1)].

References

Yann Bugeaud and Carlos Gustavo T. de A. Moreira (2011). “Sets of exact approxima-tion order by rational numbers III”. Acta Arith. 146.2, pp. 177–193. MR: 2747026(cit. on p. 748).

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Received 2017-12-13.

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