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Faculty of Science

Einstein Institute of Mathematics

MEASURES WITH UNIFORM SCALING SCENERY

Dissertation submitted by

Matan Gavish

in fulllment of the requirements for the degree of Master of Science

under the supervision of Professor Hillel Furstenberg

Submitted on 16/8/2008

Abstract. We introduce a property of measures on Euclidean space, termed 'Uniform Scaling

Scenery'. For these measures, the empirical distribution of the measure-valued signal, obtained

by rescaling around a point, is (almost everywhere) independent of the point. This property

is related to existing notions of self-similarity: we show that it is satised by the occupation

measure of a typical Brownian motion (notably 'statistically' self-similar), as well as by the

measures associated to attractors of ane iterated function systems (notably 'exactly' self-

similar). It is possible that dierent notions of self-similarity are unied under this property.

The proofs trace an intriguing connection between uniform scaling scenery and CP-Processes,

which is a class of natural, discrete-time, measure-valued Markov processes, useful in fractal

geometry.

The study of measures with uniform scaling scenery and the connection to CP-Processes were

suggested by Hillel Furstenberg, who supervised this work.

1

2 MEASURES WITH UNIFORM SCALING SCENERY

Thanks

Thanks Professors Mordechai Epstein ZL and Alexander Olevskii for unfolding the amazing

beauty of analysis: these are the fruits of their eorts.

I thank with all my heart Benny Chor (a.k.a Skipper) for his wonderful, constant support, and

for mentoring an ignorant undergraduate in asking the right questions, as well as in writing

scientic essays.

Uri, thanks for teaching me, by way of example, the three pillars of true scientic erudition:

unbound curiosity, drive for utter comprehension in all walks of science, and reaching for the

best education (globally).

This thesis has been conceived and shaped by my master's advisor, Hillel Furstenberg, who with

endless patience has given me a unique perspective on mathematics, and is the most attentive

advisor any student can hope for. Still, I learned much more than math. I wish to dedicate a

familiar verse to Hillel. During the two years I spent in Jerusalem as your student, you have

taught me its true meaning:

You have been told, O man, what is good, and what the Lord requires of you: Only to do the

right and to love goodness, and to walk humbly with your God. (Micah 6:8)

During the past year, I have spent hours on hours in the oce of Benji Weiss. Thank you so

much for your countless, indispensable teachings.

I am indebted to Noam Berger, Uri Shapira and Mike Hochman for helpful discussions.

Two years ago, Dorit Aharonov suggested that I come to Jerusalem and take a look around

the math department. Thanks for this advice! I have never seen, and probably never will, a

department of better professional and personal quality.

Finally, half this work is actually due to my wonderful wife, Ifat, without whose incredible

belief and support I would not have made it. (Thanks for accommodating my horrible character

during writing, by the way).

This work is dedicated to the loving memory of my grandmother, Klara Bender ZL, who passed

away during its preparation.

MEASURES WITH UNIFORM SCALING SCENERY 3

Contents

Abstract 1

1. Prologue 5

1.1. Outline 5

1.2. Structure of this text 5

2. Some preliminaries 6

2.1. Probability measures on compact and locally compact spaces 6

2.2. The category of measure-preserving dynamical systems. Factors. 8

2.3. The Stone-Weierstrass theorem and applications 9

2.4. The Rohlin Natural Extension 9

2.5. Two-sided Brownian motion 13

3. The empirical distribution of a signal 14

3.1. The empirical distribution of a signal taking values in a nite state space 14

3.2. Empirical distribution of a signal taking values in a compact state space 16

3.3. Generic points 18

3.4. Empirical distribution of a signal in continuous time 19

4. The uniform continuous-scale scenery property 24

4.1. Continuous-scale scenery 24

4.2. Uniform scenery 26

4.3. The Brownian occupation measure has uniform scaling scenery 28

5. CP-Systems 33

5.1. The basic idea behind CP-Systems 34


5.2. Toy example 35

5.3. The p-ary decomposition of the unit cube Q in Rd 37

5.4. Denition of a CP-System 39

5.5. Avoiding degeneracy 40

5.6. Examples of CP-Systems 42

6. Discrete scaling scenery 44

6.1. Cubes and framesets 44

6.2. Discrete scaling scenery and its empirical distribution 45

6.3. Uniform discrete scaling scenery 47

6.4. Generic points of a CP-System and uniform CP-scaling scenery 49

7. Centered scaling scenery in extended CP-systems 52

7.1. Extended CP-Systems 52

7.2. Discrete central scenery in CP-Systems 54

8. Uniform continuous-scale scenery in ergodic CP-Systems 59

9. Self-similarity and uniform scaling scenery 65

9.1. Iterated Function Systems 66

9.2. Ergodic CP-System generated by a single measure 67

9.3. Self-similar measures have uniform scaling scenery 70

10. Further Questions 70

Appendix: Extending a CP-System using the Rohlin Natural Extension 71

List of symbols 76

References 77


1. Prologue

1.1. Outline. Consider a measure θ on Rd and a point x ∈ support (θ). Use a microscope

with a continuous knob, centered on x, to expand and rescale θ by a factor of et; call the rescaled

measure θx,t. The continuous-scale scenery map t 7→ θx,t is a continuous-time signal with a

measure-valued alphabet: as such, it may or may not have a limiting empirical distribution. In

case it does, it corresponds to a continuous-time measure-valued stationary process, η [θ, x]. The

map x 7→ η [θ, x] is an interesting structure attached to θ, pertaining to its fractal behavior. of

special interest is the case where the map x 7→ η [θ, x] is θ-almost everywhere constant, namely

when θ exhibits just a single process η [θ] regardless of where we place our microscope. Our

main objective is to make precise and investigate this property of uniform continuous-scale

scenery. Do such measures exist? A typical Brownian occupation measure in Rd (d > 3) is a

natural example. What is perhaps more surprising, the classical Cantor measure is another;

This stems from an intriguing relation with a class of natural, discrete-time, measure-valued

Markov processes, called CP - (Conditional Probability) Processes. These systems (which are

due to H. Furstenberg) provide a general approach for attacking problems in fractal geometry.

We show that almost any measure in an ergodic CP-process demonstrates uniform continuous-

scale scenery, thus obtaining a large collection of measures with this property. This raises several

natural questions regarding the nature of the connection between CP-processes (which are

discrete-time, and therefore possess inherent scale) and measures with uniform scaling scenery

(a property that does not depend on scale). Finally, we combine our main result with the general

CP-system approach mentioned above to show that any self-similar measure (a natural measure

associated with the attractor of certain iterated function systems) has uniform continuous-scale

scenery. The notions of self similarity in Brownian motion (considered statistical in nature)

and of self similarity in, say, the Cantor measure (considered exact) are therefore unied under

this property.

1.2. Structure of this text. This text is organized as follows. 2 summarizes some of the

necessary measure theory, ergodic theory and probability. In 3 we discuss the abstract notion

of limiting empirical means and empirical distribution of discrete-time and continuous-time

signals, taking values in a compact space. The property of uniform continuous-scale scenery is

introduced in 4, and it is shown that the Brownian occupation measure is a non-trivial example.

The plot takes an unexpected turn in 5, which is dedicated to the seemingly unrelated subject

of a class of dynamical systems called CP-Systems. In 6, the discrete version of a scaling


scenery of a measure is dened, and the notion of discrete uniform scaling scenery is introduced.

We show that a typical measure in an ergodic CP-System has uniform discrete scaling scenery

(Theorems 6.11 and 7.13). In order to discuss continuous-scale scenery in CP-Systems, we

rst extend a CP-System to a system of measures on the entire Euclidean space; This is done

in 7. 8 is dedicated to our main result, Theorem 8.1, whereby a typical measure in an

ergodic CP-System has uniform continuous-scale scenery. In 9, we combine our main result

with the general technique oered by CP-Systems to prove that self-similar measures, which

are natural measures on the attractors of ane iterated function systems, have uniform scaling

scenery (Theorem 9.3). The appendix contains the detail construction of extended CP-Systems,

needed for 7.

2. Some preliminaries

Most of the results below are classical and are brought without proof.

2.1. Probability measures on compact and locally compact spaces. For any topological

space X, we denote by Cc (X) the set of functions X → R with compact support. This space

is always equipped with the supremum norm. When X satises two minimal conditions, the

dual Cc (X) can be identied as a space of measures on X:

Theorem 2.1. The Riesz Representation Theorem for locally compact spaces [Ru2]

Let X be a locally compact Hausdor space, and let φ : Cc (X)→ R be a positive linear functional

(namely, φf > 0 for all f > 0). Then there exists a σ-algebra G on X , which contains all the

open sets (and therefore all Borel sets in X) and there exists a unique complete measure νφ on

G, such that

(1) φf =´X f dνφ for all f ∈ Cc (X)

(2) νφK <∞ for all K ⊂ X compact

(3) For every A ∈ G we have νφ(A) = infνφ(V )

∣∣A ⊂ V , V open

(4) For every open set A, and for every A ∈ G with νφ (A) < ∞, we have νφ(A) =

supνφ(K)

∣∣K ⊂ A , K compact

Denition 2.2. A Borel measure ν on a locally compact Hausdor space X satisfying prop-

erties 2-4 above is called a Radon measure.


Theorem 2.3. A Borel probability measure on a compact metric space X is a Radon measure.

[Bil, theorem 1.1]

Suppose then that X is a locally compact Hausdor space. The Riesz representation theorem

identies the dual space of Cc (X) as the space of Radon measures on X. Recall [Ru3] that if V

is a topological vector space then the weak-* topology on its dual V ∗ is the weakest topology in

which all the functionals fx : V ∗ → R, where fx : ϕ 7→ ϕ(x), are continuous. (Equipped with the

weak-* topology, V ∗ is always a locally convex topological vector space). The space of Radon

measures on X, under the identication with Cc (X)∗, is equipped with the weak-* topology in-

duced from Cc (X). A local sub-basis at zero is given byU (f1, . . . , f` ; ε)

∣∣ fi ∈ Cc (X) , ε > 0

where

U (f1, . . . , f` ; ε) =µ ∈M (X)

∣∣ ∣∣∣∣ˆXfi dµ

∣∣∣∣ < ε , i = 1, . . . , `.

Denition 2.4. For a locally compact Hausdor space X, writeM (X) for the space of Radon

measures on X with the weak-* topology induced from Cc (X).

Lemma 2.5. Let K be a compact topological space. Let P (K) ⊂ M (K) be the subspace of

Borel probability measures in the subspace topology. Then P (K) is compact and metrizable.

Proof. First note that by Theorem 2.3, a Borel probability measure is Radon, so P (K) ⊂M (K) as sets. For C (K) 3 f > 0 dene

Ef =θ ∈M1

∣∣ ˆKf dθ > 0

= T−1

f [0,∞)

where Tf : θ 7→´K f dθ. By the denition of the weak-* topology, Tf is continuous, hence

Ef ⊂M (K) is closed. Next, dene

Epositive =θ ∈M (K)

∣∣ ˆKf dθ > 0 ∀f > 0

Enormalized =

θ ∈M (K)

∣∣ ˆK

1K dθ = 1.


Since Epositive =⋂f>0Ef , we nd that Epositive ⊂ M (K) is closed. Since 1 1K ∈ C (K) and

Enormalized = T−11K1, we nd that Enormalized ⊂M (K) is closed. Hence,

P (K) = Epositive⋂Enormalized

is closed inM (K).

Denote by B1 =f ∈ C (K)

∣∣ supK |f | < 1the unit ball in C (K). This is an open neigh-

borhood of 0. Invoking the Banach-Alaoglu theorem [Ru3, theorem 3.15], we nd that the

subset

M1 =θ ∈M (K)

∣∣ ∣∣∣∣ˆKf dθ

∣∣∣∣ 6 1 ∀ f ∈ B1

is compact inM (K). By [Ru3, theorem 3.16], M1 is also metrizable. Clearly P (K) ⊂M1, so

P (K) is a closed subset of a metrizable compact space, hence compact metrizable itself.

For a compact metric space X, the space C (X) (of continuous, real valued functions with the

supremum norm) is separable [Ru1]. We therefore obtain the following important

Corollary 2.6. (1) Let K be a compact space. Then C (P (K)) is separable.

(2) Let A ⊂ C (P (K)) is a dense set. The subalgebra generated by A over Q is a dense

R-subalgebra of C (P (K)).

The following theorem is sometimes called the Portmanteau Theorem [Bil, theorem 2.1]. It

determines continuity and discontinuity of functions of the form θ 7→ θ (A). Below, we write A

for the topological closure of some Borel set A ⊂ X,A for its interior, and ∂A = A \

A for its

boundary.

Lemma 2.7. Let X be a metric space. Fix a Borel set A ⊂ X and dene fA : P (X)→ R by

fA : θ 7→ θ (A). Then fA is continuous at a point θ if and only if θ (∂A) = 0.

2.2. The category of measure-preserving dynamical systems. Factors. Let (X,B, µ)

be a probability space, and T : X → X be a measurable, measure preserving (abbreviated m.p.)

transformation. We say that (X,B, µ, T ) is a m.p. dynamical system, or simply a m.p. system.

Let (X,B, µ, T ) and (X ′,B′, µ′, T ′) be m.p. systems, and assume that Y ∈ B and and Y ′ ∈ B′

are such that µ (X \ Y ) = µ′ (X ′ \ Y ′) = 0. Dene BY =Y ∩A

∣∣A ∈ B. This is a σ-algebraon Y , contained in B. Dene µY to be the restriction of µ to BY . Consider the measure spaces

1If we replace K by the locally compact space X, we nd that 1X /∈ Cc (X). This is precisely the reason that

P (X) fails to be compact when X is not compact.


(Y,BY , µY ) and(Y ′,B′Y ′ , µ′Y ′

). A function f : Y → Y ′ is called a morphism of the measure

spaces (X,B, µ) and (X ′,B′, µ′) and sometimes denoted2 f : X → X ′. If f : Y → Y ′ is an

injective function, it is said that f : X → X ′ is injective (mod 0); if f : Y → Y ′ is onto Y ' ,

it is said that f : X → X ′ is onto (mod 0). f : X → X ′ is called a morphism of m.p. systems

(X,B, µ, T ) and (X ′,B′, µ′, T ′) if f T ′ = T f except perhaps for a set of µ-measure zero.

Since both (X,B, µ) and (X ′,B′, µ′) are probability spaces, such f must be onto mod 0. We

then say that (X ′,B′, µ′, T ′) is a factor of (X,B, µ, T ) and call f a factor map.

2.3. The Stone-Weierstrass theorem and applications. We will make extensive use of

the following classic result [Ru1]:

Theorem. (Stone-Weiestrass Theorem, Real Version). Let X be a compact Hausdor space

and L an R- subalgebra of C(X), which contains a non-zero constant function. Then L is dense

in C(X) if and only if it separates points.

As an easy corollary, we nd a useful way to generate dense sets in product spaces:

Corollary 2.8. Let X be a compact Hausdor space L is a subalgebra of C(X), which contains a

non-zero constant function. Let I be an index set3 and consider XI =x = (xα)α∈I

∣∣∀α. xα ∈ Xin the product topology. Dene L to be the linear span (over R) of the collection of functions

x 7→∏i=0

fi(xα(i)

) ∣∣∣ ` ∈ N ; f0, . . . , f` ∈ L ; α(0), . . . , α(`) ∈ I

.

Then L is a subalgebra of C(XI), and is dense if and only if L separates the points of X.

2.4. The Rohlin Natural Extension. Let (X,B, µ, T ) be a measure preserving dynamical

system. In general, T may not be invertible (though it must be onto mod 0). We now construct

an invertible measure preserving system(X, B, µ, T

), for which (X,B, µ, T ) is a factor. The

system(X, B, µ, T

)is known as the Rohlin Natural Extension of (X,B, µ, T ), and is essentially

unique. See [Ro1] and [PY, pp.96].

The idea underlying this construction can be illustrated in simple terms as follows. Any discrete

time stationary stochastic process can be modeled by a measure preserving dynamical system:

if X = X1, X2, . . . is a stationary process taking values in the measure space (S,F), then there

2Note the abusive use of the symbol ”→ ” to denote a point function and a categorical morphism alternatively.

3We will use I ⊂ Z and I ⊂ R.


exist a measure preserving dynamical system (X,B, µ, T ) and a function f : X → S such that

the stationary process f, f T, f T 2, . . . has the same distribution as X (in the sense that

they have they same nite dimensional distributions). The system (X,B, µ, T ) along with the

function f are called the ergodic theory model of the process X. Concisely, X is the product

set∏∞i=0 S , B is the product σ-algebra

∏∞i=0F , µ is the unique measure4 on (X,B) for which

µ

(s0, s1, . . .) ∈ X∣∣ si ∈ Ai , n 6 i 6 m = P Xi ∈ Ai , n 6 i 6 m for all n 6 m ∈ N and

every An, . . . , Am ∈ F , and T is the shift on X, dened by T (s0, s1, . . .) = (s1, s2, . . .). The

function f is simply the projection f : (s0, s1 . . .) 7→ s0. Thus the construction of the ergodic

theory model of the process X can be described as looking into the future and attaching to

each state s ∈ S all possible sample paths of the form (s, s1, s2, . . .). The space of sample paths

X is equipped with a probability measure µ, which reects the probability that the various

sample paths actually occur under X.

Conversely, if (X,B, µ, T ) is a measure-preserving dynamical system and f : (X,B) → (S,F)

is measurable, dene a series of S-valued random variables on X by

Xi = f T i , i = 0, 1, 2, . . . .

Clearly, Xii∈N is an S-valued stationary process, whose ergodic theory model is (X,B, µ, T ).

Thus a stationary process can be modeled by a dynamical system together with an integrable

function and vice-versa, allowing us to use these two formulations of the same object inter-

changeably. For example, a stationary process is said to be a ergodic simply if the corresponding

dynamical system is ergodic.

To any stationary process X = X1, X2, . . . there corresponds a consistent family of nite-

dimensional probability distributions. We can use this family and the Kolmogorov consistency

theorem to dene a two-sides stationary process by X = . . . X−2, X−1, X0, X1, X2, . . . having

the same nite dimensional distributions. In the world of stationary processes, the process X

can be termed the natural extension of the process X.

Now, for a measure preserving system (X,B, µ, T ), we construct the Rohlin natural extension(X, B, µ, T

)formally as follows. Dene

X =0∏

n=−∞X = . . .×X × . . .×X ×X ,

4Existence and uniqueness of the measure µ follow from the Kolmogorov consistency theorem [D].


denote the elements of X by x = (. . . , x−n, . . . , x−1, x0) and let πi : X → X be the projection

on the coordinate i, that is,

πi : (. . . , x−n, . . . , x−1, x0) 7→ xi .

For any measurable sets A0, . . . A−k ∈ B, denote the induced cylinder set by

CA0,...A−k =

(. . . , x−1, x0) ∈ X∣∣xi ∈ Ai , i = 0, . . .− k

.

Let B be the product σ-algebra induced from B on X, that is, the σ-algebra generated by the

projections, πi0i=−∞. Dene µ on the cylinder sets by

(2.1) µ(CA0,...A−k

)= µ

(0⋂

i=−kT−i−kAi

).

For instance, this means that µ(CA0,A−1,A−2

)= µ

(T−2A0 ∩ T−1A−1 ∩A−2

). It is easy to ver-

ify directly that this is a consistent family of measures on cylinders, so by the Kolmogorov consis-

tency theorem [D] there exists a unique measure (also denoted by µ), which satises (2.1) for any

cylinder set. Note that the support of µ is precisely the set of sequences (. . . , x−n, . . . , x−1, x0),

for which T (xi) = xi+1. Now, dene T : X → X by T (. . . , x−2, x−1, x0) = (. . . , x−1, x0, Tx0),

namely (Tx)i

=

xi+1 i < 0

Tx0 i = 0.

Note that T is invertible on the support of µ, with inverse

T−1 : (. . . , x−2, x−1, x0) 7→ (. . . , x−3, x−2, x−1) .

Indeed, if (. . . , x−1, x0) ∈ X satises T (xi) = xi+1, then

T T−1 (. . . , x−1, x0) = T−1 T (. . . , x−1, x0) = (. . . , x−1, x0) .

This means that T is invertible µ- mod 0. Also, for each cylinder set we have

T−1CA0,...A−k =

(. . . , x−1, x0) ∈ X∣∣Tx0 ∈ A0, x0 ∈ A1, . . . , x−k+1 ∈ A−k

=

= CT−1A0⋂A−1,A−2,...A−k .

Hence, the preimage of any cylinder by T is a cylinder, and since the cylinders generate B,we nd that T is measurable. To show that µ is T -invariant, suppose that CA0,...A−k is some


cylinder set. We have

µT−1CA0,...A−k = µCT−1A0⋂A−1,A−2,...A−k =

= µ

(T−k

(T−1A0 ∩A−1

)∩

[ −1⋂i=−k+1

T−k−iAi−1

])=

= µ

(T−k−1A0 ∩

[0⋂

i=−k+1

T−k−iAi−1

])=

= µ

T−k−1A0 ∩

−1⋂j=−k

T−1(T−k−jAj

) =

= µT−1

T−kA0 ∩

−1⋂j=−k

T−k−jAj

=

= µT−1

0⋂j=−k

T−k−jAj

= µ

0⋂j=−k

T−k−jAj

=

= µCA0,...A−k ,

where the next to last equality is due to the fact that µ is T -invariant. We have thus dened(X, B, µ, T

)and established that it is a measure preserving system, and that T is invertible µ-

mod 0. The fact that (X,B, µ, T ) is a factor of(X, B, µ, T

)is immediate: dene π : X → X

by π : x→ x0. Since for any A ∈ B we have π−1A = CA and in fact µπ−1A = µCA = µA, π is

measurable and measure preserving. Finally, by the denition of T the diagram

X

π

T // X

π

X

T // X

commutes, so that (X,B, µ, T ) is indeed a factor.

We mention two additional properties of the natural extension:

(1) It can be shown [Ro1] that an additional condition renders the natural extension unique

up to an isomorphism of measure preserving systems.

(2) The natural extension of an ergodic system is ergodic.


2.5. Two-sided Brownian motion. Fix d and let B(1)s and B

(2)s be independent, standard

Brownian motions in Rd. The standard two-sided Brownian motion Bs in Rd is dened by

Bs =

B(1)s s > 0

B(2)−s s 6 0

.

Note that P B0 = 0 = 1 so that almost surely, the sample path Bs is an innite, continuous

path through the origin in Rd. Below, we abbreviate standard, two-sided Brownian motion

simply by BM. We will abuse notation by denoting both the random curve ω 7→ Bs (ω), and a

given realization (or sample path) Bs (ω), simply by Bs.

For a radius r > 0, dene Tr = sups ‖Bs‖ = r − infs ‖Bs‖ = r. This is an upper bound on

the time Bs spends inside the ball of radius r around the origin. We call a BM Bs transient if for

all r > 0, we have Tr <∞ almost surely. Since for d > 2 almost surely both sups∥∥∥B(1)

s

∥∥∥ = 1

and sups∥∥∥B(2)

s

∥∥∥ = 1are nite for any r > 0, it follows that BM in d > 2 is transient.

Proposition. (Brownian Scaling) Let Bs be a standard (one-sided) Brownian Motion in R and

let 0 < a ∈ R. Dene Bs = 1√aBas. Then Bs is a standard Brownian Motion.

Proof. Recall that if Xt an almost surely continuous Gaussian process with EXs = 0 and

Cov (Xt, Xs) = t ∧ s, then Xt is a standard Brownian motion [KS, PMö]. Clearly, Bs

is almost surely continuous with EBs = 0. But Cov(Bt, Bs

)= Cov

(1√aBat,

1√aBas

)=

1aCov (Bat, Bas) = 1

a (at ∧ as) = t ∧ s.

Corollary 2.9. If Bs is a BM in Rd, then so is Bs = 1√aBas.

This is known as Brownian Scaling and suggests studying the transformation f(s) 7→ 1√af (as)

on the space of continuous paths through the origin in Rd from a dynamical system perspective.

Dene

Ω =f : R→ Rd

∣∣ f(0) = 0 , f continuous.

For each n ∈ N, dene |f |n = supx∈[−n,n] |f(x)|. Then |·|n∞n=1 is a separating family of semi-

norms, which turns Ω into a Fréchet space5[Ru3]. In this topology, fn → f if and only if (fn)

converges to f uniformly on any compact set K ∈ R. Let B denote the corresponding Borel

5As this topology is induced by a complete, separable metric, Ω is in fact a Polish space. This is convenient,

as any complete probability measure turns it into a standard probability space.


σ-algebra on Ω. The distribution of a BM process induces a Borel measure on (Ω,B), which

we call the Wiener measure and denote by µw.

Denition 2.10. For t ∈ R, dene a transformation Ψt : Ω→ Ω by (Ψtf) (s) = et2 f(e−ts

).

Proposition 2.11. Ψt is a measure-preserving ow on (Ω,B, µw).

Proof. The semigroup law Ψt+r = Ψt Ψr is clearly satised for any t, r ∈ R. By Corollary 2.9,

Ψt is measure preserving. Proving that is Ψ : R× Ω→ Ω is Borel measurable is routine.

An interesting, yet somewhat less famous fact is that [Fi1]

Proposition 2.12. Ψt is a mixing ow on (Ω,B, µw).

In fact, it is shown in [Fi1] that the ow Ψtt∈R is (the) Bernoulli ow of innite entropy on

Ω.

3. The empirical distribution of a signal

3.1. The empirical distribution of a signal taking values in a nite state space. Let

Λ be a nite alphabet, and consider a signal taking values in Λ, namely x = (x0, x1, . . .) ∈ ΛN.

We x ` ∈ N and a nite word of length ` + 1, denoted a = a0a1, . . . a` ∈ Λ`+1. One may

wonder about the asymptotic average rate of appearance of the word a in the signal x, formally

dened as

(3.1) limN→∞

1N

N−1∑n=0

1xn+`n =a (x) = lim

N→∞

1N

N−1∑n=0

∏i=0

1xn+i=ai (x) .

Obviously, for each word a ∈ Λ∗, this limit may or may not exist.

Denition 3.1. If the limit (3.1) exists for any ` ∈ N and any nite word a ∈ Λ`+1, we say

that the signal x has limiting empirical means.

If the signal x has limiting empirical means, we obtain a consistent family of probability mea-

sures on the nite dimensional products, Λn∞n=1, by setting

ηx (a) = limN→∞

1N

N−1∑n=0

1xn+`n =a (x) for a ∈ Λ`+1 .


The Kolmogorov consistency theorem implies that ηx extends uniquely to a shift-invariant Borel

probability measure on ΛN (equipped with the product topology), or in other words, a Λ-valued

stationary process in discrete time. We call ηx the empirical distribution of x. 6

Thus, any signal x ∈ ΛN with limiting empirical means induces an empirical distribution ηx,

which is a probability measure on ΛN. From the dynamical point of view, in this case x is a

generic point for the system(ΛN, ηx, σ

)where σ is the shift.

While this denition does explain the origin of the terms limiting empirical means and em-

pirical distribution, it does not extend naturally to the case of innite state-space Λ. As we

will ultimately be interested in the case where Λ is a space of measures, we need to introduce

a condition, equivalent to that of Denition 3.1, which lends itself naturally to the case of Λ

innite.

Proposition 3.2. Let Λ be a nite set and consider a signal x ∈ ΛN. Denote by σ : ΛN → ΛN

the shift onto ΛN. Then the following are equivalent:

(1) x has empirical means in the sense of Denition 3.1.

(2) The sequence of functionals ΦN : C(ΛN) → R dened by ΦN : g 7→ 1

N

∑N−1n=0 g σn (x)

converges weakly to the functional Φ : g 7→´

ΛN g d ηx.

(3) The sequence of probability measures ηx,n = 1N

∑N−1n=0 δσnx converges to ηx in the w*-

topology induced by C(ΛN) on P (ΛN).

(4) For any ` and any continuous functions f0, . . . , f` ∈ C (Λ), the limit

limN→∞

1N

N−1∑n=0

∏i=0

fi (xn+i)

exists.

Proof. (2⇔3) is just the denition of the w*-topology on P(ΛN). We now prove 1⇔2⇔4. Note

that since Λ is compact, so is the Tychono product ΛN. (1⇒2) Let A be the linear R-span ofg : ΛN → R

∣∣ g (y) =∏i=0

1yi=ai for a = a0a1, . . . a` ∈ Λ`+1

6Due to the plethora of measures that will appear in our later discussion, we use the term distribution

for ηx, rather than measure. Throughout this work, any measure on a space of measures is referred to as

distribution.


(as before, we denote y = (y0, y1, . . .) ∈ ΛN). Clearly, A ⊂ C(ΛN) is a subalgebra, which sepa-

rates points in ΛN and contains a nontrivial constant function. Hence by the Stone-Weierstrass

theoremA is a dense subalgebra. Observe that the functionals ΦN are continuous and ‖ΦN‖ 6 1

for all N . By assumption, ΦNg → Φg for all g ∈ A, a dense subspace. A standard argument

now implies that in fact, ΦNg → Φg for all g ∈ C(ΛN), as required. (2⇒1) is obvious. (4⇒2)

Dene A′ to be the linear R-span ofg : ΛN → R

∣∣ g (y) =∏i=0

fi (yi) for ` ∈ N and f0, . . . f` ∈ C (Λ)

.

Again by the Stone-Weierstrass theorem, A′ ⊂ C(ΛN) is a dense subalgebra. By assumption,

limN→∞ΦNg exists for all g ∈ A'. Again, since the norms of ΦN are bounded, convergence on

the dense subspace A′ implies that ΦN converges weakly on C(ΛN). For g ∈ C

(ΛN), denote

the limit limN→∞ΦNg by Φg. Clearly, Φ is a linear functional on A′ with ‖Φ‖ 6 1. It remains

to show that Φg =´

ΛN g d ηx. Since A ⊂ A′, all limits of the form (3.1) exist, and by the

denition of ηx, limN→∞ΦNg = Φg =´

ΛN g d ηx for any g ∈ A, which is a dense subspace of

C(ΛN). It follows that Φg =

´ΛN g d ηx for all g ∈ C

(ΛN). (2⇒4) is obvious.

3.2. Empirical distribution of a signal taking values in a compact state space. Let

us now replace the nite alphabet Λ by a compact topological space X. Consider a signal

x = (x0, x1, . . .) ∈ XN. Recall from 2.1 that if (θn)∞n=1 ⊂ P (X) is a sequence of probability

measures on X, then θn → θ ∈ P (X) in the weak-* topology induced by C (X) if and only if

θnA → θA for any Borel set A with θ (∂A) = 0. Hence, conditions (1) and (2) of Proposition

3.2 are unfortunately no longer equivalent, since indicator functions no longer oer a simple

criterion for the existence of empirical means. The most natural approach for the empirical

distribution of a signal taking values in a compact state space is thus through continuous

functions, namely condition (2) of Proposition 3.2. The practical denition is given by the

following generalization of condition (4) Proposition 3.2.

Denition 3.3. Let X be a compact space and x = (x0, x1, . . .) ∈ XN. If for any ` and any

continuous functions f0, . . . , f` ∈ C (X), the limit

limN→∞

1N

N−1∑n=0

∏i=0

fi (xn+i)

exists, we say that the signal x has limiting empirical means.


This denition implies the existence of an empirical distribution. The proof of the next propo-

sition is identical to that of Proposition 3.2:

Proposition 3.4. Let X be a compact space and consider a signal x ∈ XN. Denote by σ the

shift on XN. The following are equivalent:


(2) The sequence of functionals ΦN : C(XN)→ R dened by ΦN : g 7→ 1

N

∑N−1n=0 g σn (x)

converges weakly to some positive functional Φ on C(XN) with Φ (1) = 1.

If x has empirical means, then the limit functional Φ corresponds, by the Riesz representation

theorem, to a measure ηx such that 1N

∑N−1n=0 g σn (x) →

´XN g dηx for all g ∈ C

(XN). The

measure ηx on XN is called the empirical distribution corresponding to x. This obviously agrees

with the denition of the empirical measure for the case of X nite.

We will make extensive use of the following lemma, which allows us to check the condition in

Denition 3.3 on a smaller (and in practice, countable) family of functions.

Lemma 3.5. Let X be a compact space and x = (x0, x1, . . .) ∈ XN. Let A ⊂ C (X) be a

subalgebra which contains a nonzero constant function. Assume that A is dense, or that it

separates points in X. If for any ` and any continuous functions f0, . . . , f` ∈ C (X), the limit

limN→∞

1N

N−1∑n=0

∏i=0

fi (xn+i)

exists, then x has limiting empirical means. Moreover, the empirical distribution ηx is uniquely

determined by the values ˆ

XN

[∏i=0

fi (yi)

]dηx(y) .

Proof. Dene A to be the R-linear span of functions of the form y 7→∏`i=0 fi (yi) for ` ∈ N

and f0, . . . , f` ∈ A, namely

A =

g(y) =r∑j=1

cj∏i=0

fji (yi)∣∣ `, r ∈ N , fji ∈ A , cj ∈ R

so that A ⊂ C

(XN) is a subalgebra. By Corollary 2.8, A is dense in C

(XN). Since the sequence

of functionals ΦN : C(XN)→ R dened by ΦN : g 7→ 1

N

∑N−1n=0 g σn (x) converges weakly on


A, and since we have ‖ΦN‖ 6 1 for all N , we nd that ΦN converges weakly on C(XN). Denote

the limit functional by Φ. It is easy to check that Φ is positive with ‖Φ‖ 6 1 and Φ (1) = 1.

Thus, by Proposition 3.4, x has limiting empirical means. As Φ is determined by its values

on the dense subspace A, we nd that the empirical distribution ηx, which is by denition the

measure corresponding to the limit functional Φ, is uniquely determined by the values

Φ

(y 7→

∏i=0

fi (yi)

)=ˆ

XN

[∏i=0

fi (yi)

]dηx(y) .

3.3. Generic points. Let X be a compact metric space. As we have seen, any individual

signal (sequence) x taking values in X may have limiting empirical means, which dene an

empirical distribution ηx on XN. Suppose now that B and µ are the Borel σ-algebra and a

Borel probability measure on X, and that T : X → X is a measurable (though not necessarily

continuous) map preserving µ. For x ∈ X, the signal(x, Tx, T 2x, . . .

)is said to be generated by

the stationary source (X,B, µ, T ). As we now show, the Birkho ergodic theorem implies that

µ-almost every x ∈ X, the signal(x, Tx, T 2x, . . .

)has limiting empirical means. A set G ⊂ X

with µG = 1 where this holds is a generic set for the source (X,B, µ, T ), and its elements are

called generic points.

Proposition 3.6. Let (X,B, µ, T ) be as stated. Then there exists a measurable set G ⊂ X with

µG = 1, such that for any x ∈ G the signal x =(x, Tx, T 2x, . . .

)has limiting empirical means.

Moreover, if (X,B, µ, T ) is ergodic, then G can be chosen such that ηx, the empirical measure

corresponding to x =(x, Tx, T 2x, . . .

), is identical for all x ∈ G.

Proof. The remark above Corollary 2.6 implies that C (X) is separable. Let A ⊂ C (X) be a

countable, sense subset. We can assume that the constant function 1 ∈ A. Enumerate A by

A = gi∞i=1. For any ` ∈ N and any `+ 1-tuple (k0, . . . , k`) ∈ N`+1, dene

g(k0,...,k`) =∏i=0

gki Ti

so that g(k0,...,k`) ∈ L1 (X,B, µ). By Birkho's ergodic theorem, there exist a set G(k0,...,k`) ⊂ X

with µG(k0,...,k`) = 1 such that for all x ∈ G(k0,...,k`) , we have

limN→∞

1N

N−1∑n=0

[g(k0,...,k`) T

nx]

= limN→∞

1N

N−1∑n=0

[∏i=0

gki(Tn+ix

)]


exists. Now, dene

G =⋂`∈N

⋂(k0,...,k`)∈N`+1

G(k0,...,k`) .

Since this intersection is countable, µG = 1. But for any x ∈ G, the following condition holds:

for all ` ∈ N and for any nite set gk0 , . . . , gk` ∈ A, the limit

limN→∞

1N

N−1∑n=0

[∏i=0

gki Tn+ix

]exists. This is just condition the in Lemma 3.5, so for any x ∈ G, the signal x =

(x, Tx, T 2x, . . .

)has empirical means. Now, if in addition the system (X,B, µ, T ) is ergodic, then for all x ∈ G,we have

ˆ

XN

[∏i=0

gki(y)

]dηx(y) = lim

N→∞

1N

N−1∑n=0

[∏i=0

gki(Tn+ix

)]

= limN→∞

1N

N−1∑n=0

[g(k0,...,k`) T

nx]

=ˆ

X

g(k0,...,k`)(x′) dµ(x′) .

Since the values of´XN

[∏`i=0 gki(y)

]dηx(y) are independent of the choice of x ∈ G, Lemma

3.5 implies that the empirical distribution of the signal x =(x, Tx, T 2x, . . .

)is identical for all

x ∈ G.

3.4. Empirical distribution of a signal in continuous time. Since we will be interested

in measure-valued signals in continuous time, as well as in discrete time, we need to generalize

the notion of limiting empirical means and empirical distribution to the case of a signal in

continuous time, taking values in a compact space. Replacing the shifts in discrete time with

shifts in continuous time, we get the following analogy to Denition 3.3:

Denition 3.7. Let X be a compact space, and let x = (xt)t∈R ∈ XR be a signal in continuous

time taking values in X. If for any `, any f0, . . . , f` ∈ C (X), and every (τ0, . . . , τ`) ∈ (R)` the

limit

limT→∞

1T

T

t=0

[∏i=0

fi (xt+τi)

]dt

exists, we say that x has (well-dened) empirical means.


As in the case of a discrete-time signal, the condition in the above denition is often easy

to verify, but strong enough to imply the existence of empirical distribution. The following

proposition is analogous to Proposition 3.4:

Proposition 3.8. Let X be a compact space and consider a signal x ∈ XR. Denote by στ :

XR → XR (τ ∈ R ) the group of shift operators, (σtx)τ = xτ+t. The following are equivalent:


(2) The net of functionals ΦT : C(XR)→ R dened by

(3.2) ΦT : g 7−→ 1T

t=Tˆ

t=0

g σt (x) dt

converges weakly, as T →∞, to some positive functional Φ on C(XR) with Φ (1) = 1.

Proof. (1⇒2) First note that the map σ(·) (x) : R → C(XR) dened by σ(·) (x) : t 7→ σt (x) is

measurable. Since XR is compact, any C(XR) is bounded, so that the integral in (2) in fact

exists. Dene A′ to be the linear R-span ofg : XR → R

∣∣∣∣∣ g (y) =∏i=0

fi (yτi) for ` ∈ N , τ0, . . . , τ` ∈ (R)` and f0, . . . f` ∈ C (X)

.

by Corollary 2.8 (which followed from the Stone-Weierstrass theorem), A′ ⊂ C(XR) is a dense

subalgebra. By assumption, limT→∞ΦT g exists (in the sense of nets) for all g ∈ A'. As the

norms of ΦT are bounded, convergence on the dense subspace A′ implies that ΦT converges

weakly on C(XR). For g ∈ C (XR), denote the limit limT→∞ΦT g by Φg. Clearly, Φ is a positive

linear functional on A′ with ‖Φ‖ 6 1 and Φ (1) = 1. (2⇒1) Choose `, f0, . . . , f` ∈ C (X), and

(τ0, . . . , τ`) ∈ (R)`. Dene g (y) =∏`i=0 fi (yτi). By assumption, the limit

limT→∞

1T

t=Tˆ

t=0

g σt (x) dt = limT→∞

1T

T

t=0

[∏i=0

fi (xt+τi)

]dt

exists.

If x has empirical means, then the limit functional Φ corresponds, by the Riesz representation

theorem, to a Borel measure (with respect to the product topology) ηxon XR+

such that

1T

t=Tˆ

t=0

g σt (x) dt→ˆ

XR

g dηx


for all g ∈ C(XR). The measure ηx is called the empirical measure, or empirical distribution,

corresponding to x.

We now note that expressions of the form´ Tt=0

[∏`i=0 fi (xt+τi)

]dt are continuous in the time-

shifts, τi`i=0. In fact, we have

Proposition 3.9. Let X be a compact space, and let x = (xt)t>0 ∈ XR be a signal taking values

in X. For ` ∈ N, g = (g0, . . . , g`) ∈ C (X)`+1 and τ = (τ0, . . . , τ`) ∈ (R)`+1, set ψg;τ : XR → Rby

ψg;τ : x 7−→T

t=0

[∏i=0

gi (xt+τi)

]dt

(the dependence on T is implicit in the notation ψg;τ ). Then ψg;τ is Lipschitz in τ , namely there

exists a constant C = C (g, `) such that for τ = (τ0, . . . , τ`) ∈ (R)`+1 and λ = (λ0, . . . , λ`) ∈(R)`+1 , we have supx∈XR

∣∣ψg;τ (x)− ψg;λ (x)∣∣ < C

∥∥τ − λ∥∥1.

Proof. Let us demonstrate the idea for the special case ` = 0. In this case

ψg;τ (x) =

T

t=0

g (xt+τ ) dt

for some g ∈ C (X) and τ ∈ R. For λ ∈ R, a change of variables gives

ψg;λ (x) =

t=Tˆ

t=0

g(xt+τ+(λ−τ)

)dt =

s=T+(λ−τ)ˆ

s=(λ−τ)

g (xs+τ ) ds =

=

s=Tˆ

s=0

g (xs+τ ) ds+

s=0ˆ

s=(λ−τ)

g (xs+τ ) ds+

s=T+(λ−τ)ˆ

s=T

g (xs+τ ) ds

and hence

(3.3) |ψg;τ (x)− ψg;λ (x)| =

∣∣∣∣∣∣∣s=0ˆ

s=(λ−τ)

g (xs+τ ) ds

∣∣∣∣∣∣∣+∣∣∣∣∣∣∣s=T+(λ−τ)ˆ

s=T

g (xs+τ ) ds

∣∣∣∣∣∣∣ 6 2·|λ− τ |·supx|g| .

Generally, for arbitrary ` we take a similar approach. Choose θ ∈ P(Qx,0

), so

∏i=0

gi (xt+τi)−∏i=0

gi (xt+λi) =

=∑k=0

k−1∏i=0

gi (xt+λi) ·∏i=k+1

gi (xt+τi) · [gk (xt+τk)− gk (xt+λk)]


where we use the convention that the product over the empty set is 1. Now, denoting M =

(maxi supx |gi|), we get

∣∣ψg;τ (x)− ψg;λ (x)∣∣ =

∣∣∣∣∣∣t=Tˆ

t=0

∏i=0

gi (xt+τi) dt−t=Tˆ

t=0

∏i=0

gi (xt+λi) dt

∣∣∣∣∣∣ 66 M `−1 ·

∑k=0

∣∣∣∣∣∣t=Tˆ

t=0

gk (xt+τk)−t=Tˆ

t=0

gk (xt+λk)

∣∣∣∣∣∣ .We can now use the estimate (3.3) for each of the terms to obtain

∣∣ψg;τ (x)− ψg;λ (x)∣∣ 6 M `−1 ·

∑k=0

[2 · |λk − τk| · sup

x|gk|]6 2 ·M ` ·

∥∥λ− τ∥∥1

as required.

We will make extensive use of the following lemma, which is analogous to Lemma 3.5. It allows

us to verify existence of the limit limT→∞1T

´ Tt=0


]dt for functions fi from a

smaller (and in practice, countable) family of functions, and for rational times τi.

Lemma 3.10. Let X be a compact space and x = (xt)t ∈ XR . Let A ⊂ C (X) be a subalgebra

which contains a nonzero constant function. Assume that A is dense, or that it separates points

in X. If for any ` ∈ N, any continuous functions f0, . . . , f` ∈ A and every (τ0, . . . , τ`) ∈ (Q)`+1

the limit

limT→∞

1T

T

t=0

[∏i=0

fi (xt+τi)

]dt

exists, then x has empirical means. Moreover, the empirical measure ηx is uniquely determined

by the values ˆ

XR

[∏i=0

fi (yτi)

]dηx(y) .

Proof. Choose ` ∈ N, functions f0, . . . , f` ∈ A and ` + 1-tuple τ = (τ0, . . . , τ`) ∈ (R)`. Let

us show that the limit limT→∞1T

´ Tt=0


]dt exists using Cauchy's criterion. By

Proposition 3.9 there exists a constant C such that for any τ , λ ∈ (R)`+1 we have∣∣∣∣∣∣T

t=0

[∏i=0

gi (xt+τi)

]dt −

T

t=0

[∏i=0

gi (xt+λi)

]dt

∣∣∣∣∣∣ < C ‖τ − λ‖1 .


Now, let ε > 0 and take λ = (λ0, . . . , λ`) ∈ (Q)`+1 with∥∥τ − λ∥∥

1< ε. Since by assumption

the limit limT→∞1T

´ Tt=0

[∏`i=0 fi (xt+λi)

]dt exists, take T0 > C such that for all S, T > T0 we

have ∣∣∣∣∣∣ 1T

T

t=0

[∏i=0

fi (xt+λi)

]− 1S

S

t=0

[∏i=0

fi (xt+λi)

]∣∣∣∣∣∣ < ε

and therefore for all S, T > T0∣∣∣∣∣∣ 1T

T

t=0

[∏i=0

fi (xt+τi)

]− 1S

S

t=0

[∏i=0

fi (xt+τi)

]∣∣∣∣∣∣ 66

1T

∣∣∣∣∣∣T

t=0

[∏i=0

fi (xt+τi)

]−

T

t=0

[∏i=0

fi (xt+λi)

]∣∣∣∣∣∣+

+

∣∣∣∣∣∣ 1T

T

t=0

[∏i=0

fi (xt+λi)

]− 1S

S

t=0

[∏i=0

fi (xt+λi)

]∣∣∣∣∣∣+

+1S

∣∣∣∣∣∣S

t=0

[∏i=0

fi (xt+λi)

]−

S

t=0

[∏i=0

fi (xt+τi)

]∣∣∣∣∣∣ 66C∥∥τ − λ∥∥

1

T+ ε+

C∥∥τ − λ∥∥

1

S< 3ε .

Setting h(T ) = 1T

´ Tt=0


]dt, we thus have that for all ε > 0 there exists T0 such

that for all S, T > T0,

|h(T )− h(S)| < ε .

Hence the limit

limT→∞

h(T ) = limT→∞

1T

T

t=0

[∏i=0

fi (xt+τi)

]dt

exists.

Now, dene A to be the R-linear span of functions of the form y 7→∏`i=0 fi (yτi) for ` ∈ N and

f0, . . . , f` ∈ A, namely

A =

g : XR → R

∣∣∣∣∣ g (y) =∏i=0

fi (yτi) for ` ∈ N , τ0, . . . , τ` ∈ (R)` and f0, . . . f` ∈ C (X)

so that A ⊂ C(XR) is a subalgebra, and by Corollary 2.8, dense in C

(XR). Since the net of

functionals ΦT : C(XR) → R dened by in (3.2) converges weakly on A, and since we have

‖ΦT ‖ 6 1 for all T , we nd that the net ΦT converges weakly on C(XR). Denote the limit


functional by Φ. It is easy to check that Φ is positive with ‖Φ‖ 6 1 and Φ (1) = 1. Thus,

by Proposition (3.8), x has empirical means. Now, as Φ is determined by its values on the

dense subspace A, we nd that the empirical measure ηx, which is by denition the measure

corresponding to the limit functional Φ, is uniquely determined by the values

Φ

(y 7→

∏i=0

fi (yτi)

)=ˆ

XR

[∏i=0

fi (yτi)

]dηx(y) .

4. The uniform continuous-scale scenery property

Consider a measure θ on Rd and a point x ∈ support (θ). We now make precise the notion of

a microscope with a continuous knob, centered on x, which allows us to expand and rescale

θ by a factor of et and normalize it, thus obtaining the rescaled measure θx,t. We consider

the scaling scenery t 7→ θx,t as a continuous-time signal with a measure-valued alphabet,

and appeal to 3.4 for the notion of limiting empirical means of this signal. If it has limiting

empirical means, we denote the corresponding empirical distribution by η[FRx , θ

]. We next

consider the map x 7→ η[FRx , θ

]. Where dened, it is an interesting structure attached to

θ. We identify similar empirical distribution attached to dierent points using an equivalence

relation ∼ and ask whether there exist measures, for which the map x 7→ η[FRx , θ

]is θ-a.e

constant (mod ∼). This amounts to asking for measures that exhibit just a single empirical

distribution of the scaling scenery. This property, which we term uniform continuous-scale

scenery is the main subject of this text.

4.1. Continuous-scale scenery . How to give formal meaning to the notion of rescaling a

measure θ (dened on Rd) around a point x ∈ Rd? While the simplest approach would be

to rst move x to the origin (compare the denition of Tangential Measure at [PrMö]), the

following approach will suit our later needs better. We use a sequence of nested, concentric

cubes decreasing to x. Since we think of each cube as a frame, the entire sequence is called a

frameset. A frameset is naturally equipped with a sequence of homotheties, each mapping a

small cube onto the basic cube - the cube of side-length 1 around x. Rescaling θ by a factor

of e−t around x is then achieved by pushing θ forward by the homothety that maps the small

cube of side-length e−t onto the basic cube. In order to prevent mass from escaping to innity,

we then normalize θ on the basic cube.


Denition 4.1. Let x = (x1, . . . , xd) ∈ Rd. The family of cubes FRx =

Qx,t

t∈R, where

Qx,t =d∏i=1

[xi −

12e−t , xi +

12e−t],

is called the continuous frameset 7 around x. Also, abbreviate and write Qx for

Qx,0 =d∏i=1

[xi −

12, xi +

12

].

Let H⟨Qx,t , Qx

⟩: Rd → Rd denote the unique homothety inducing a bijection of Qx,t onto

Qx. Explicitly,

H⟨Qx,t , Qx

⟩(y) = ety + (1− et)x .

Some notation. We denote by M(Rd)the set of Radon measures on Rd with the weak-*

topology (recall Denition 2.4), and by P (Qx) the set of probability measures on Qx for some

x ∈ Rd (recall Lemma 2.5). Finally, write |θ| for the support of θ.

We now consider the sequence of measures, obtained from rescaling a xed measure θ around

a point x in its support.

Denition 4.2. Fix x ∈ Rd and let FRx =

Qx,t

t∈R be the continuous frameset around x.

Suppose that θ ∈ M(Rd)is a Radon measure on Rd with x in its support. The sequence8

θx,tt∈R of measures on Qx, where

θx,t =H⟨Qx,t , Qx

⟩ θ

θ(Qx,t

) ∣∣∣Qx

will be called the continuous-scale scenery of θ around x. (Note that since x ∈ |θ|, the denom-

inator never vanishes.)

Remark 4.3. Compare our denition to the notion of scenery ow in [Fi2, Fi1] and the notion

of tangential measure in [PrMö].

Denition 4.4. Let θ ∈ M(Rd)and x ∈ |θ|. Consider x =

θx,tt∈R, the F

Rx -scenery of θ,

as a P (Qx)-valued signal in continuous time (recall from Lemma 2.5 that P (Qx) is a compact

space). If the signal x has limiting empirical means (in the sense of Denition 3.7), we say

that θ has FRx -limiting empirical means. The corresponding empirical distribution ηx, which

7This peculiar choice of notation is meant to conform with more general notation to be introduced later.

8Actually, a net.


is a probability measure on P (Qx)R (or, equivalently, a P (Qx)-valued stationary process in

continuous time) is called the FRx -empirical distribution of θ and denoted by η

[FRx , θ

].

Thus the scaling scenery of θ around x is the data acquired by a microscope with a continuous

knob placed on θ at the point x. Analyzing this measure-valued signal, we may nd that is

has denite asymptotical statistical behavior, namely that is has limiting empirical means. In

this case, the empirical distribution η[FRx , θ

]encapsulates this statistical behavior.

4.2. Uniform scenery. We thus nd an additional structure attached to a measure θ ∈M(Rd): to any point x ∈ |θ|, such that θ has FR

x -limiting empirical means, there corre-

sponds the P (Qx)-valued, continuous-time stationary process η[FRx , θ

].

Of special interest is the case when the mapping x 7→ η[FRx , θ

]is θ-a.e dened and, in

some sense to be dened, constant. We need a way to compare the processes η[FRx , θ

]and

η[FRy , θ

]for x 6= y ∈ |θ|, although they have dierent state-spaces: one is P (Qx) -valued and

the other is P (Qy) - valued.

Denition 4.5. Let x, y ∈ Rd and let ηx (resp. ηy) be distributions of continuous-time,

P (Qx) (resp. P (Qy))-valued stationary processes9. Suppose that whenever the process θt is

distributed ηx, the process H 〈Qx , Qy〉 θt is distributed 10 ηy. We then say that ηx and ηy

are equivalent and write ηx ∼ ηy.

The relation ∼ is easily seen to be an equivalence relation. Let us write η∼ for the equivalence

class of a distribution η. The following easy observation will help us compare the empirical

distributions of θ ∈M(Rd)at dierent points:

Lemma 4.6. Let θ ∈ M(Rd)and x, y ∈ |θ|. Let

(θx,t)t∈R and

(θy,t)t∈R be the continuous-

scale sceneries of θ around x and y. If the P (Q0)-valued signals(H 〈Qx , Q0〉 θx,t

)t∈R and(

H 〈Qy , Q0〉 θy,t)t∈R both have limiting empirical means with the same empirical distribution,

then:

(1) θ has FRx and FR

y limiting empirical means, and

(2) η[FRx , θ

]∼ η

[FRy , θ

].

9in other words, ηx is a shift-invariant probability measure on P (Qx)R

10this amounts to the statement that H 〈Qx , Qy〉 induces an isomorphism of the systems (P (Qx) , ηx, σ) and

(P (Qy) , ηy, σ) where σ is the shift ow, σs : θt 7→ θs+t


Now that we are able to compare the empirical distribution of θ around dierent points, we

come to the notion of uniform continuous-scale scenery, which is the main subject of this work.

Suppose that θ ∈M(Rd)has FR

x -limiting empirical means around θ-a.e x ∈ Q. Observe thatinstead of the structure x 7→ η

[FRx , θ

], we should in fact attach the structure x 7→ η

[FRx , θ

]∼

to θ. Consider the case when there is only one equivalence class above θ, namely when the

map x 7→ η[FRx , θ

]∼ is constant on Q, θ-almost everywhere11. We call this property uniform

continuous-scale scenery.

Denition 4.7. Let θ ∈M(Rd)be a Radon measure. We say that it has uniform continuous-

scale scenery (or uniform FR-scenery) if for θ-almost every x ∈ Q, θ has FRx - limiting

empirical means, and the mapping x 7→ µ[FRx , θ

]∼ is θ-a.e. constant on Q. In this case, any

representative of the equivalence class that is the (almost everywhere) constant value of this

mapping is called the uniform empirical distribution, and denoted by η[FR, θ

].

Note that if θ has uniform FRx -scenery, then the uniform empirical distribution η

[FR, θ

]can

be chosen to be a probability measure on P (Q0)R.

The property of uniform continuous-scale scenery can be described in simple terms as follows.

We have a measure θ ∈ Rd. We take a volunteer bug with a mathematical inclination, provide

him with a microscope and a computer, and lead him, blind-folded, to some point x ∈ Q ∩ |θ|.The bug is then granted eyesight and is required to determine his whereabouts (the point x)

without moving around. He may use his microscope to gather local data on θ around x and

process asymptotical statistics of this data with his computer. If θ has uniform continuous-

scale scenery, then θ-almost every point x ∈ Q will look the same, rendering our bug's attempts

futile.

Example. The Lebesgue measure on Rd:

Let λ be the Lebesgue measure on Rd. This is the easiest example of a measure with uniform

scaling scenery: for any x ∈ Q (indeed for any x ∈ Rd) the FRx -scaling scenery of λ is

λx,t ≡ λ∣∣Qx

. Therefore , for any x ∈ Q, λ has FRx - limiting empirical means, and the empirical

distribution is a dirac delta, supported on the sample path with a constant value λ∣∣Qx

. The

representative of the constant value of the map x 7→ η[FRx , λ

]∼ can be chosen to be a dirac

delta, supported on the sample path with constant value λ∣∣Q0.

11Since we are interested in small-scale structure of measures, we only consider the map x 7→ η[FRx , θ

]∼

on the cube Q = [0, 1]d and not on the entire space Rd.


Combining Lemma 3.10 and Lemma 4.6, we obtain a criterion for uniform continuous-scale

property, which will be the one we actually use:

Lemma 4.8. Let θ ∈ M(Rd)and assume that A ⊂ C (P (Q0)) is a dense subalgebra with

1Q0 ∈ A. If there exists a set H ⊂ Q with θ (Q \H) = 0 such that for any ` ∈ N, any

continuous functions f0, . . . , f` ∈ A and every (τ0, . . . , τ`) ∈ (Q)`+1 the limit

limT→∞

1T

T

t=0

[∏i=0

fi

(H 〈Qx , Q0〉 θx,t+τi

)]dt

exists and is identical for each x ∈ H, then θ has uniform continuous-scale scenery and the

uniform empirical distribution η = η[FR, θ

]is uniquely determined by the values

ˆ

P(Q0)R

[∏i=0

fi (yτi)

]dη(y) .

4.3. The Brownian occupation measure has uniform scaling scenery. A natural ex-

ample for a collection of measures with uniform continuous-scale scenery is generated by a

transient Brownian motion.

Let us recall the notation and results of 2.5, and again abbreviate two-sided, standard Brow-

nian Motion through the origin in Rd by BM. Fix d > 3, so that BM is transient. Denote by

(Ω,B, µw) the probability space of continuous paths through the origin in Rd in the topology

of convergence on compact sets, its corresponding Borel σ-algebra, and the Wiener measure

induced by a BM process. We now dene a random measure on Rd, called the BM occupation

measure.

For a given BM sample path12 Bs, dene a Borel measure on θB on Rd by

θB (A) =ˆ

Rd

1A (Bs) dλ(s) for A ∈ B

where λ is the Lebesgue measure on R. Note that if the sample path Bs is considered as a

Borel function t 3 R B7−→ Bs ∈ Rd , then θB is just the push-forward measure λ B. If Bs

is considered as a random curve, then θB is a random measure on Rd. It is easy to verify the

following

12Since the variable t is used to denote scaling, we write Bs for a Brownian path and s for the path parameter.


Proposition 4.9. (i) The support of θB is the path Bs itself. (ii) Almost surely, θB is a Radon

measure on Rd.

Recall from 2.5 that (Ψtf) (t) = et2 f(e−ts

)denes a mixing ow on (Ω,B, µw). For our

current purpose, will be more convenient to use the mixing ow (Ψtf) (t) = etf(e−2ts

). Recall

our notation Q0 =[−1

2 ,12

]d.

For some xed BM path B ∈ Ω, what is the relation between the continuous-scale scenery

(θB)0,t of the occupation measure θB around the origin, and the occupation measures θΨtB

corresponding to the path ΨtB? The answer is given in the following

Lemma 4.10. Dene ρ : Ω→ C (P (Q0)) by

ρ : B 7−→ 1θBQ0

· θB∣∣∣Q0

restricted to Q0. Then

(θB)0,t = ρ (ΨtB) .

Proof. For A ⊂ Q0, we have

θΨtB (A) = λs ∈ R

∣∣ etBe−2ts ∈ A

= e2t · λs∣∣Bs ∈ e−tA = e2t · θB

(e−tA

)and since H

⟨Q0 , Q

0,t

⟩A = e−tA, the denition of the scaling continuous-scale scenery of θB

yields

(θB)0,t A =

(H⟨Q0,t , Q0

⟩ θB

)A

θB

(Q0,t

) =

=θB(H⟨Q0 , Q

0,t

⟩A)

θB

(H⟨Q0 , Q0,t

⟩Q0

) =

=λ(B−1

[e−tA

])λ (B−1 [e−tQ0])

=

=θB(e−tA

)θB (e−tQ0)

=θΨtB (A)θΨtB (Q0)

.

Now, set

ρ : B 7−→ 1θBQ0

· θB

so that ρ maps a BM path into the corresponding occupation measure restricted to Q0, nor-

malized so as to yield a probability measure. Thus

(θB)0,t = ρ (ΨtB) ,


as required.

We are now ready to prove that for µw-almost any BM path B, the corresponding occupation

measure θB has uniform continuous-scale scenery. In fact, we will prove that the uniform

empirical distribution is shared by (almost any) BM path! Several ideas in the proof will play

a central role in the sequel.

Theorem 4.11. There is a set of BM paths G ⊂ Ω with µwG = 1 such that for every B ∈ G,θB has uniform scaling scenery. Moreover, the mapping B 7→ η

[FR, θB

]is constant on G,

where η[FR, θB

]is the uniform empirical distribution of the measure θB.

Proof. Let us start with the scaling scenery around the origin of Rd, x = 0. We rst show

that there exists G ⊂ Ω, with µwG = 1, such that for any B ∈ G, the occupation measure

θB has FR0 -limiting empirical means, and in fact, that the empirical distribution µ

[FRx , θB

]is identical for any B ∈ G. This will follow from a variation on the theme of generic points,

introduced in 3.3.

Corollary 2.6 implies that C (P (Q0)) is separable. Let A = gi∞i=1 be a countable, dense

set. For ` ∈ N, ` + 1-tuple k = (k0, . . . , k`) ∈ N`+1 and ` + 1-tuple τ (τ0, . . . τ`) ∈ Q dene

g(k ; τ) : Ω→ R by

g(k ; τ) =∏i=0

gki ρ Ψτi

where ρ : Ω → C (P (Q0)) is as in Lemma 4.10. As g(k ; τ) is clearly measurable and bounded,

we have g(k ; τ) ∈ L1 (Ω,B, µw). We now apply the Birkho-Khinchin (pointwise) ergodic

theorem for ergodic ows [CFS, pp.11 theorem 1] to the mixing ow (Ψt)t∈R, and obtain a set

G(k ; τ) ⊂ Ω with µwG(k ; τ) = 1 such that for all B ∈ G(k ; τ) , we have

limT→∞

1T

t=Tˆ

t=0

[g(k ; τ) Ψt (B)

]dt =

ˆ

Ω

g(k ; τ) dµw.

Note that this limit does not depend on B. Now, let

G =⋂`∈N

⋂k∈N`+1

⋂τ∈Q

G(k ; τ)


so that µwG = 1. All this means that for any B ∈ G we have the following: for any ` ∈ N, anycontinuous functions gk0 , . . . , gk` ∈ A and any every (τ0, . . . , τ`) ∈ (Q)` the limit

limT→∞

1T

t=Tˆ

t=0

[g(k ; τ) Ψt (B)

]dt = lim

T→∞

1T

t=Tˆ

t=0

[∏i=0

gki ρ Ψt+τi (B)

]dt =

= limT→∞

1T

t=Tˆ

t=0

[∏i=0

gki

[(θB)0,t+τi

]]dt

exists and does not depend on B (the last equality above follows from Lemma 4.10). Thus by

Lemma 3.10, for any B ∈ G, the occupation measure θB has FR0 - limiting empirical means,

and moreover, the empirical distribution of θB at x = 0, namely η[FR

0 , θB], is identical for

any B ∈ G.

Since the support of θB is justBs∣∣ s ∈ R ⊂ Rd, in order to consider the scaling scenery

around points in |θB| other than 0, we actually need to discuss the FRBa

-scenery of θB for any

a ∈ R. Fix, then, some a ∈ R and dene an operator Sa : Ω → Ω by (SaB)t = Bt+a − Ba.(Note the fundamental property of Brownian motion, that Sa preserves the Wiener measure

µw). By the translation invariance of the Lebesgue measure λ we have

θSaB (A) = λs∣∣Bs+a −Bs ∈ A = λ

s∣∣Bs ∈ A+Ba

= θB (A+Ba)

Now, FRBa

=Q0,t +Ba

t∈R. Since obviously

H⟨QBa , Q

Ba,t

⟩A = H

⟨Q0 , Q

0,t

⟩(A−Ba) +Ba ,

we can write

(θB)Ba,t A =

(H⟨QBa,t , QBa

⟩ θB

)A

θB

(QBa,t

) =

=θB

(H⟨QBa , Q

Ba,t

⟩A)

θB

(H⟨QBa , Q

Ba,t

⟩QBa

) =

=θB(H⟨Q0 , Q

0,t

⟩(A−Ba) +Ba

)θB

(H⟨Q0 , Q0,t

⟩(Q0) +Ba

) =

=θSaB

(H⟨Q0 , Q

0,t

⟩(A−Ba)

)θSaB

(H⟨Q0 , Q0,t

⟩Q0

) =

= (θSaB)0,t (A−Ba)


so (θB)Ba,t A = (θSaB)0,t (A−Ba). In other words, we have

(θSaB)0,t (A) = (θB)Ba,t (A+Ba) =(H 〈QBa , Q0〉 (θB)Ba,t

)A ,

that is,

(4.1) (θSaB)0,t = H 〈QBa , Q0〉 (θB)Ba,t .

Fix some a ∈ R. Note that Sa is continuous13. This means that if B ∈ Ω is such that SaB ∈ G,then we have for any `, k and τ as above

ˆ

Ω

g(k ; τ) dµw = limT→∞

1T

t=Tˆ

t=0

g(k ; τ) Ψt (SaB) =

= limT→∞

1T

t=Tˆ

t=0

∏i=0

gki (θSaB)0,t+τi dt =

= limT→∞

1T

t=Tˆ

t=0

∏i=0

gki

[H 〈QBa , Q0〉 (θB)Ba,t+τi

]dt

where the last equality follows from (4.1). Dene Ga = S−1a G and note that since µw is Sa-

invariant, µwGa = 1.

Finally, consider the product measure space (Ω, µw)×(R, λ). Set G =

(B, a) ∈ Ω× R∣∣B ∈ Ga.

Obviously ˆ

R

ˆΩ

1([Ω×R]\G)dµw

dλ =ˆ

R

0 dλ = 0 ,

but since both µw and λ are σ-nite, it follows from Fubini's theorem [Par2, pp. 180] that

0 =ˆ

R

ˆΩ

1([Ω×R]\G)dµw

dλ =ˆ

Ω×R

1([Ω×R]\G) d (µw × λ) =ˆ

Ω

ˆR

1([Ω×R]\G)dλ

dµw .This means that there exists a set G with µwG = 1 with the following property: for any B ∈ G,and for λ-almost any a ∈ R, we have B ∈ Ga. As θB is absolutely-continuous with respect to

λ, this implies that for any B ∈ G, and for θB-almost any x ∈ Rd, the following holds: if `, k

and τ are as above, then

(4.2) limT→∞

1T

t=Tˆ

t=0

∏i=0

gki

[H 〈QBa , Q0〉 (θB)x,t+τi

]dt =

ˆ

Ω

g(k ; τ) dµw

13We expand on the continuity of measure translation later. See (7.4) and Lemma 7.10.


and in particular, the limit on the left exists and does not depend on B or on x.

Choose some B ∈ G. In particular, the limits (4.2) does not depend on x, so we can invoke our

simple criterion for uniform continuous-scale scenery, Lemma 4.8, to obtain that the Brownian

occupation measure θB has uniform continuous-scale scenery. Denote its uniform empirical

distribution by η[FR, θB

], and recall that it is determined by all limits of the form (4.2).

However, as the limits in (4.2) are also independent of the choice of B ∈ G, we nd that the

mapping B 7→ η[FR, θB

]is in fact constant on G.

Remarks:

(1) Consider a BM path B for which θB has uniform continuous-scale scenery. Note that

while the map x 7→ η[FRx , θB

]is θB-almost everywhere constant on Q, it is not

everywhere constant. For example, if x is a local maximum, in one of the axes, of the

Brownian path, then all measures in the continuous-scale scenery of θB around x are

supported on half the cube Qx. Therefore, the empirical distribution of θB around a

local maximum is markedly dierent from the uniform empirical distribution14.

(2) Note that the phenomenon exhibited by the Brownian occupation measure is stronger

than uniform continuous-scale scenery. Not only do we nd a single (continuous-time,

P (Q0)-valued) stationary process above some measure θB, but we discovered an en-

semble of measures with uniform continuous-scale scenery,θB∣∣B ∈ G, all sharing a

single uniform empirical distribution.

(3) The zero set of a Brownian Motion can yield a similar and equally natural example for

measures with uniform continuous-scale scenery, which we do not develop here.

5. CP-Systems

The property of uniform continuous-scale scenery appears to be connected to self-similarity

in some sense. Does the Cantor measure, for instance, have uniform scaling scenery? The

answer is yes. This stems from a connection with a class of natural, discrete-time, measure-

valued Markov processes, called CP- (Conditional Probability) Processes, which were originally

introduced to provide a useful approach for attacking problems in fractal geometry. Our main

result roughly states that an ergodic CP-process generates many non-trivial measures with

uniform scaling scenery.

14Thanks Noam Berger for this observation.


5.1. The basic idea behind CP-Systems. Let Q ⊂ Rd be a unit cube, Q = [0, 1]d. Choose

a grid-size parameter p ∈ N and divide each of the d axis into p equal subintervals of length

1p . This induces a decomposition of Q into pd subcubes of side-length 1

p , each indexed by

the choice of a subinterval in each axis. For each λ ∈ Λ = 0, . . . , p− 1d write Qλ for the

corresponding subcube. Iterating this procedure, we get a decomposition of Q into p2d subcubes

of side length 1p2, indexed by Λ× Λ. Continuing this way, we get the p-ary decomposition of

Q discussed in 5.3.

Consider the space P (Q) of probability measures on Q 15. We use the symbol θ to denote

elements of this space. Let us regard P (Q) as a state-space of a Markov process. Write θλ

for the measure θ∣∣Qλ

, stretched, translated and normalized to yield a probability measure on

Q. The decomposition Q =⊎λ∈ΛQ

λ suggests a natural transition function: from the state

θ ∈ P (Q) we make the transition to θ → θλ with probability θ(Qλ). In terms of a Markov

operator P : C (P (Q))→ C (P (Q)), this is expressed as

(Pf) (θ) =∑λ∈Λ

θ(Qλ)f(θλ).

Ergodic theorems now enter the discussion in a very natural way: if Ω ⊂ P (Q) is a closed

subset, which is invariant under the transition θ 7→ θλ for all λ, then it is a compact state space

for the above Markov process. Therefore [D] there exist ergodic stationary measures on Ω.

For technical reasons, it is desirable to pass from the formalism of stationary Markov processes

to the formalism of measure preserving dynamical systems. The standard way of achieving

this is to consider the product space Ω × Ω × . . . with the shift operator and a probability

measure, induced by the transition probabilities. An element (θ1, θ2, . . .) of the product space

has the interpretation of an initial state θ1 along with its path in the future. However, [F1, F2]

suggested a simpler alternative of attaching the future path to an initial state θ ∈ P (Q), and

thus admitting a dynamical system formalism: The sequence of transitions θ 7→ θλ1 7→ θλ1λ2 7→. . . corresponds to the choice of a sequence of subcubes Q, Qλ1 , Qλ2 , . . ., decreasing to some

point x ∈ Q. It is easily veried that the probability law of the choice of the point x is non

other than θ itself. Therefore, instead of considering the state θ along with a the future path(θ, θλ1 , θλ1λ2 , . . .

)chosen Ω × Ω × . . . with the probability law induced on the product space

by the transition probabilities, we consider the couple (θ, x) where x ∈ Q is chosen according

15As noted in 2.1, the weak-* topology induced by the space of C (Q) turns it into a compact, metric space.


to θ. When x is not located on the face of some subcube of the p-ary decomposition, the map

(λ1, λ2, . . .)→ Q dened by

(λ1, λ2, . . .) 7→∞∑n=1

λnpn

is injective, so that the choice of x according to θ determines the future path and has the same

probability law.

This leads to the denition of a CP-System, which is a dynamical system formulation of our

natural Markov process on P (Q). The measurable space is a subspace of P (Q) × Q. The

transformation maps(θ ,∑∞

n=1λnpn

)to(θλ1 ,

∑∞n=1

λn+1

pn

). Finally, if the initial distribution

of the Markov process is ν, then the distribution on P (Q) × Q will describe the choice of a

measure θ according to ν, followed by a choice of a point x according to θ.

The notions of a CP-Process and a CP-System were suggested implicitly in [F1] and dened in

[F2]. See [F2, FW] for applications of the CP-Process notion and an equivalent denition using

trees. See [LPP1, LPP2, LPP3] and [LP, chapter 16] for interesting applications to random

walks on Galton-Watson trees.

5.2. Toy example. Here is a simple example, intended to elucidate the notions of CP-System

and uniform CP-scenery to follow.

Consider the following sequence of tilings of the real unit interval interval, [0, 1]. The n-th tiling

consists of 2n intervals of length 2−n, denoted by I(n)a1...an , where

I(n)a1...an =

[n∑i=1

ai2i,

n∑i=1

ai2i

+12n

)for 11 . . . 1 6= a1, . . . an ∈ 0, 1n

are just the reals with prex a1, . . . an in their binary representation, and I(n)11...1 =

[1− 1

2n , 1].

For completeness, denote I(0)∅ = [0, 1]. Let µ be a Borel probability measure on [0, 1], and

suppose that x is not a dyadic point, and that it lies in the support of µ, which we denote

by |µ|. For each n, Let I(n)x be the unique interval among

I

(n)a1...an

containing x. Since

x ∈⋂∞n=1 I

(n)x and µI

(n)x > 0 for each n, we can zoom-in around x (using the intervals I

(n)x )

and ask about the behavior of the measure µ as we do. Formally, we look at the the sequence

of conditioned and rescaled measures on [0, 1], dened by

(5.1) µ(n)x A =

1

µI(n)x

µ(ϕ−1A

),


where n > 0, A ⊂ [0, 1], and ϕ : I(n)x [0, 1] is the bijective ane map. This is the dyadic scal-

ing scenery we see when we zoom-in around x, each time taking the dyadic interval containing

x in the next level.

Let us construct a random probability measure ν on [0, 1], a typical instance of which will

be shown to demonstrate peculiar behavior of this scaling scenery. Consider Ω =

13 ,

23

N, and

let us map a sequence x = (x1, x2, . . .) ∈ Ω to a measure ν = ν (x) as follows. We simplify the

notation below by writing, for a ∈ 0, 1 and x ∈

13 ,

23

x〈a〉 = (x)(1−a) + (1− x)a =

x a = 0

1− x a = 1.

Fix x = (x1, x2, . . .) ∈ Ω and set ν I(1)a = x

〈a〉1 for a = 0, 1. Explicitly,

ν

[0,

12

)= x1

so that

ν

[12, 1]

= 1− x1 .

Now, use recursion to set ν on I(n)a1...an for a1 . . . an ∈ 0, 1n by

ν I(n)a1...an−1an = x〈an〉n · ν I(n−1)

a1...an−1

which reads easier as

ν I(n)a1...an−10 = (xn) · ν I(n−1)

a1...an−1

ν I(n)a1...an−11 = (1− xn) · ν I(n−1)

a1...an−1

This determines the measure ν (x). Recall (5.1) and observe that for any y ∈ [0, 1] and

ν = ν (x1, x2, . . .) ,

ν (x1, x2, . . .)(n)y = ν (xn+1, xn+2, . . .)b2nyc .

To study the sequenceν

(n)y

∞n=1

of probability measures on [0, 1] (the dyadic scaling scenery

of ν around y), let us x m ∈ N and a1 . . . am ∈ 0, 1m, and consider the numerical sequenceν

(n)y I

(m)a1...am

∞n=0

. For n = 0, ν(0)y = ν so we have ν

(0)y I

(m)a1...am = νI

(m)a1...am =

∏mi=1

(x〈ai〉i

).

Similarly, regardless of y we have

ν(n)y I(m)

a1...am =m∏i=1

(x〈ai〉n+i

).


To construct our random measure, let µ be the product measure induced on Ω by a process of

identically distributed, independent, fair coin tosses. For a random sequence x in the probability

space (Ω, µ), we obtain a random measure ν (x). The system (Ω, µ, σ), where σ is the shift, is

ergodic and Ω is compact. Hence, µ-almost surely, a point (x1, x2, . . .) ∈ Ω is a generic point.

But if (x1, x2, . . .) is a generic point, for any I(m)a1...am we have

limN→∞

1N

N−1∑n=0

ν(n)y I(m)

a1...am = limN→∞

1N

N−1∑n=0

m∏i=1

(x〈ai〉n+i

)=

= limN→∞

1N

N−1∑n=0

f (σnx) =

=ˆ

Ω

f dµ

where f(x) =∏mi=1

(x〈ai〉i

). Thus if x is generic and ν = ν (x) is the corresponding measure,

the limit limN→∞1N

∑N−1n=0 ν

(n)y I

(m)a1...am exists and is independent of x and y. This argument

actually implies that the same is true for any limit of the form

limN→∞

1N

N−1∑n=0

[∏i=0

ν(n+i)y Ii

]

where I0, . . . I` are any intervals of the form I(m)a1...am . Since the dyadic intervals generate the

Borel sets in [0, 1], it follows from the Stone-Weierstrass Theorem that if x is a generic point of

(Ω, µ, σ) then ν (x) has a peculiar property: for any point y ∈ [0, 1], the sequence of measuresν

(n)y

∞n=1

has limiting empirical means in the sense of Denition 3.3, that is, for any f ∈

C(P ([0, 1])N

)the limit 1

N

∑N−1n=0 f

(ν

(n)y , ν

(n+1)y , . . .

)exists. In fact, not only is this limit

independent of y, it is independent of the choice of the generic point x: for µ-almost every

instance ν (x) of our random measure, ν has discrete uniform scaling scenery with a single

empirical distribution.

5.3. The p-ary decomposition of the unit cube Q in Rd. We adopt the notation of the

original denitions [F2]. Fix a dimension d > 1, and letQ denote the unit cube of Rd, Q = [0, 1]d


. Choose an integer p > 2, which we shall call the grid-size parameter, and dene a partition

J0 =[0 ,

1p

)J1 =

[1p,

2p

)...

Jp−2 =[p− 2p

,p− 1p

)Jp−1 =

[p− 1p

, 1]

of [0, 1] to intervals, all but the last of which are half-closed and half-open. A partition of

Q is dened as follows: Let Λ denote the product set 0, 1, . . . , p− 1d. For each (vector)

λ = (i1, i2, . . . , id) ∈ Λ, the set Qλ = Ji1 × Ji2 × . . .× Jid is a cube in Q, and

Q =⋃λ∈Λ

Qλ

is clearly a partition of Q to disjoint cubes. For x ∈ Q, there is a unique λ (x) ∈ Λ with

x ∈ Qλ(x).

In order to construct a multi-scale partition of Q, this construction is iterated as follows.

Let ψλ denote the restriction to Qλ of the map t 7→ pt − λ (note that here, t ∈ Q and

λ ∈ Λ are vectors). Then ψλ(Qλ) is a partially open subcube of Q with side length 1, unless

λ = (p−1, p−1, . . . , p−1), in which case ψλ(Qλ) = Q. For word of length ` and λ1 . . . λ` ∈ Λ∗,

subcubes Qλ1,λ2,...,λ` are now dened inductively by

Qλ1λ2 = ψ−1λ1Qλ2

Qλ1λ2λ3 = ψ−1λ1Qλ2λ3

...

Qλ1λ2...λ` = ψ−1λ1Qλ2λ3...λ`


so the cube Qλ1λ2...λ` has side length p−` and Qλ1λ2...λ`+1⊂ Qλ1λ2...λ` . Also, if λii∈N are

such that x ∈⋂`>1Qλ1λ2...λ` , then

x =∞∑n=1

λnpn

.

Here as before, each λi is a vector in Rd. Note that this innite p-ary expansion of points

x ∈ Q is not always unique.

For a word of length `, w = λ1λ2 . . . λ` ∈ Λ∗, denote Qλ1,...,λ` by Qw and the length ` of w by

`(w). It is now possible to generalize the notation Qλ(x): For each x ∈ Q and ` ∈ N, the niteword w ∈ Λ∗ of length ` with x ∈ Qw is uniquely determined, as the intervals Ji are pairwise

disjoint. For simplicity, write Q`(x) for this Qw. Thus, Q itself can be denoted by Q∅, where

∅ ∈ Λ∗ is the empty word.

5.4. Denition of a CP-System. Note that in the context of CP-Systems, the symbol θ

stands for a measure on Q, rather than on Rd. Recall that P (Q) denotes the set of probability

measures on Q ⊂ Rd with the weak-* topology induced by C (Q). For θ ∈ P (Q), denote the

support of θ by |θ| . Dene

Φ =

(θ, x)∣∣ ∀n. θ (Qn(x)) > 0

⊂ P (Q)×Q

and equip Φ with the subspace topology induced by the product topology on P (Q)×Q. Clearly,Φ is closed in P (Q)×Q, hence a compact topological space.

Dene a transformation T : Φ→ Φ by

(5.2) T (θ, x) =

(ψλ(x)

(θ|Q1(x)

)θ (Q1(x))

, px− λ(x)

)

where ψλ and λ(x) are as dened in 5.3. To show that the range is Φ, we must show that for

all n, ψλ(x)

(θ|Q1(x)

)(Qn(px− λ)) > 0. Indeed, x n and let λ ∈ Λ and w ∈ Λn be such that

Qn+1(x) = Qλw. Then Qn(px− λ) = Qw, hence

ψλ(x)

(θ|Q1(x)

)(Qn(px− λ)) = ψλ(x)

(θ|Q1(x)

)(Qw) =

= θ(ψ−1λ Qw) = θ(Qλw) = θ(Qn+1(x)) > 0 .


It is therefore justied to write T : Φ → Φ. Note that T is not, in general, a continuous

transformation 16.

Denition 5.1. A measure µ on P(Q) × Q will be called adapted if there is a measure ν on

P(Q) such that dµ(θ, x) = dθ(x) dν(θ), i.e

(5.3)

ˆ

Φ

f(θ, x) dµ(θ, x) =ˆ

P(Q)

ˆQ

f(θ, x) dθ(x)

dν(θ)

Denition 5.2. Let Φ and T be as above, and let µ be a measure on Φ. (Φ, T, µ) is called a

CP- (Conditional Probability-) System if µ is an adapted, T -invariant measure on Φ.

In the sequel, we will be mostly interested in ergodic CP-Systems, namely in the case where

the measure-preserving system (Φ, T, µ) is ergodic.

Proposition 5.3. Suppose that (Φ, T, µ) is a CP-system, and µ is adapted to ν. If an event

occurs for µ-almost every (θ, x), then it occurs for ν-almost every θ ∈ P (Q), and for every

such θ, it occurs for θ-almost every x ∈ Q.

Proof. Let A ⊂ Φ be an event of full µ-measure. By Fubini's theorem [Par2, pp. 180],ˆ

Φ

1A (θ, x) dµ =ˆP(Q)

(ˆQ

1A (θ, x) dθ)dν = 1 .

Since ν is a probability measure, and the function f(x) =´Q 1A (θ, x) dθ is bounded by 1, it

must be constant with value 1 for ν-almost every θ. For each θ in this subset of P (Q) with full

ν-measure, a similar argument then asserts that 1A (θ, x) must be constant with value 1 on a

subset of Q having full θ-measure.

5.5. Avoiding degeneracy. For practical reasons that will become clear in the Appendix, we

would like to avoid the degenerate case of a CP-System in Rd that lives on faces with linear

dimensions < d of the cube Q ⊂ Rd. We now show that this degeneracy can be eectively

avoided in an ergodic CP-System, by showing that such a system lives on a cube of well-

dened linear dimension. Therefore, if an ergodic CP-System on Q ⊂ Rd is degenerate, we canobtain a non-degenerate ergodic CP-System by restricting to the correct face of Q, which is

the basic cube of some d′ < d dimensional subspace of Rd.

16A method similar to the one used in Lemma 7.12 can be used to show that it is Borel measurable.


Denition 5.4. Let (Φ, T, µ) be a CP-System on Q ⊂ Rd. Dene ∂Φ =

(θ, x) ∈ Φ∣∣x ∈ ∂Q.

We say the the CP-System (Φ, T, µ) is degenerate if µ (∂Φ) > 0, and non-degenerate if µ (∂Φ) = 0.

Denition 5.5. Let I = i1, . . . , ik ⊂ 1, . . . , d with 1 6 i1 < . . . < ik 6 d be a subset of

1, . . . , d. Let 0, 1k 3 w = (w1, . . . , wk) be a binary word of length d′. The set Q(I,w) =(x1, . . . , xd) ∈ Q

∣∣xin = win , n = 1 . . . kis called the (I, w)-face of

∏di=1 [0, 1] = Q ⊂ Rd.

Note that under the identication of the ane subspace(x1, . . . , xd) ∈ Rd

∣∣xin = win , n = 1 . . . k

of Rd with Rd−k, the (I, w) face of Q is the basic cube of Rd−k . Similarly, we dene the

(I, w)-face of a subcube Qw of Q (for w ∈ Λ∗).

Proposition 5.6. Let (Φ, T, µ) be an ergodic CP-System on Q ⊂ Rd. Then there exists I =

i1, . . . , ik ⊂ 1, . . . , d and 0, 1k 3 w = (w1, . . . , wk) such that

(1) µ

(θ, x) ∈ Φ∣∣x ∈ the (I, w) face of Q

(2) By identifying the ane subspace

(x1, . . . , xd) ∈ Rd

∣∣xin = win , n = 1 . . . kof Rd with

Rd−k, identifying the (I, w) face of Q with the basic cube Q′ of Rd−k, and restricting

Φ in the obvious way to a CP-System on Q′, we obtain an isomorphic, non-degenerate

ergodic CP-System.

Proof. Choose some 1 6 i 6 d and ε ∈ 0, 1, and dene

(5.4) A(n)i,ε =

(θ, x) ∈ Φ

∣∣x ∈ the (i , ε) face of Qn(x).

Observe that A(n)i,ε ⊂ A

(n+1)i,ε and that in fact

(5.5) T−1A(n)i,ε = A

(n+1)i,ε .

Since A(n)i,ε ⊂ T−1A

(n)i,ε ergodicity implies that we have either µA

(n)i,ε = 0 or µA

(n)i,ε = 1. If

µA(n)i,ε = 1 for some n, it follows from (5.5) that µA

(n)i,ε = 1 for all n, and in particular that

µA(0)i,ε = 1. Similarly, if µA

(n)i,ε = 0 for some n, then µA

(n)i,ε = 0 for all n, and in particular

µA(0)i,ε = 1. For each 1 6 i 6 d, we thus have the mutually exclusive alternatives

µA(0)i,0 = 0 , µA

(0)i,1 = 1

µA(0)i,0 = 1 , µA

(0)i,1 = 0

µA(0)i,0 = 0 , µA

(0)i,1 = 0 .


Now denote by I = i1, . . . , ik ⊂ 1, . . . , d with 1 6 i1 < . . . < ik 6 d the indices i where

µA(0)i,0 = 0 , µA(0)

i,1 = 1 or µA(0)i,1 = 1 , µA(0)

i,0 = 0. Dene w = w1, . . . , wk by A(0)i,wj

= 1 for

j = 1, . . . , k. It follows that if µ is adapted to ν ∈ P (P (Q)) then ν-almost every measure

θ ∈ P (Q) is supported on the (I, w) face of Q. Restricting every measure θ ∈ P (Q) to

the face Q(I,w) and identifying this face with the basic cube of Rd−k, we get an an ergodic

non-degenerate CP-System, which is clearly isomorphic to (Φ, T, µ).

When dealing with ergodic CP-System on Q ⊂ Rd, this theorem allows us to assume that

the dimension d is chosen such that the system is non-degenerate. Henceforth, an ergodic

CP-System will always be assumed to be non-degenerate. Non-degeneracy is important for the

transition from an ergodic CP-System to an extended ergodic CP-System, to be described in

7.

5.6. Examples of CP-Systems.

(1) The Cantor measure CP-System

Let ν be the classical (ternary) Cantor measure on [0, 1]. Observe that it is invariant

under the transformation T of a CP-System on Q ⊂ R1 with grid-size parameter p = 3.

Dene δν ∈ P (P (Q)) the dirac measure on ν, and let µν be the corresponding adapted

measure on Φ. It is easy to see that support of µν is just

(ν, x)

∣∣x ∈ |ν|(|ν| is the classical Cantor set in [0, 1]) and that in fact, µν = δν×ν. Choose a bijectionϕ : 0, 2N → |ν| such that

ϕ : (a1, a2, . . .) 7→∞∑n=1

an2n

and note that T (ν , ϕ(a1, a2, . . .)) = (ν , ϕ(a2, a3, . . .)), that is, T acts on the second

coordinate as the shift. Furnish 0, 2N with the Bernoulli shift(

12 ,

12

)structure and

dene f : 0, 2N → Φ by f : (a1, a2, . . .) 7→ (ν , ϕ(a1, a2, . . .)). This is clearly an

isomorphism of measure-preserving systems. Thus T preserves µν , and in fact ergodic,

so that (Φ, µν , T ) is an ergodic CP-System. Observe that the situation is so simple for

the choice of grid-size parameter p = 2.


(2) The Stationary Delta system

This example is adapted from [F2]. Fix a grid-size parameter p ∈ N and denote

as before Λ = 0, 1, . . . , p− 1d. Let X1, X2, . . . be a stationary, ergodic stochastic

process taking values in Λ. We use a sequence model the process, that is, we choose

a probability measure ν on the product space ΛN =x = (x1, x2, . . .)

∣∣xi ∈ Λsuch

that ν is invariant to the shift σ on ΛN, the system(ΛN, ν, σ

)is ergodic, and nite

dimensional distributions of the process X1, X2, . . . agree with the measure ν of the

corresponding cylinder sets of ΛN. In particular, if π : ΛN → Λ is the projection on the

rst coordinate, then Xn is distributed like π σn. To avoid a degenerate case, assume

that

(5.6) νx∣∣x(i)

n = p− 1 ∀n

= 0 for i = 1 . . . d

where x(i)n is the i-th coordinate of the vector xn ∈ Λ = 0, 1, . . . , p− 1d. Next, dene

a mapping ΛN → QN by x 7→ (z1, z2, . . .), where

zn(x) =∞∑k=1

xn+k

pk∈ Q .

Note that the assumption (5.6), we have zn(x) ∈ Qxn+1,xn+2,...,xn+kfor each k and ν-

almost every. x ∈ ΛN. In this event, for each k we have Qk (z0) = Qx1,x2,...,xk , hence the

Dirac delta measure δz0 satises the condition (QX1,X2,...,Xk) = 1, so δZ0 (Qn (Z0)) > 0.

It follows that for ν-a.e. x ∈ ΛN, we have(δz0(x), z0(x)

)∈ Φ. This allows us to dene

a mapping φ : ΛN → Φ by xφ7−→(δz0(x), z0(x)

). Since φ is dened ν-a.e., it is dened

in the measure-theoretical sense. Let us turn φ into a morphism of measure-preserving

systems. Dene a measure µ on Φ by the push-forward, µ = φ ν. Thus φ is by

denition measure-preserving. Now, note that as T (δz0 , z0) = (δz1 , z1), we actually

have φ σ = T φ so µ is T -invariant and φ :(ΛN, ν, σ

)→ (Φ, µ, T ) a morphism of

measure-preserving systems. This implies that (Φ, µ, T ) is a CP-system, and as a factor

of the ergodic system(ΛN, ν, σ

), it is in fact an ergodic CP-System. The condition (5.6)

guarantees that it is non-degenerate.

(3) The toy example

Let ν be the distribution (that is, the probability measure) dened on P (Q) in our

toy example from 5.2. Let µ be the corresponding adapted measure on Φ. The Markov

process described there is deterministic. It is easily seen that the CP-Process (Φ, µ, T )

is isomorphic to a Bernoulli shift, and therefore ergodic.


6. Discrete scaling scenery

Suppose that (Φ, µ, T ) is a CP-System and that (θn, xn) = Tn (θ, x) is the orbit of a point

(θ, x) ∈ Φ. Recall from 4.1 that the continuous-scale scenery of a measure θ about a point x

was obtained by inating θ∣∣Qx,t

to Qx (the cube of side length 1 around x). In an analogous

manner, the sequence (θn)∞n=1 is obtained by inating the restriction θ on a sequence of

cubes, decreasing to x. It resembles a discrete version of the continuous-scale scenery from 4.1;

However, the cubes used are not concentric. In this chapter we suggest a unied treatment

of discrete scaling scenery, obtained by using an arbitrary sequence of cubes decreasing to a

point. This is the subject of 6.1 and 6.2. We then consider the discrete scaling scenery as

a (discrete-time, measure-valued) signal and use the notion of limiting empirical distribution

from 3.2 to attach (when possible) a discrete-time, measure-valued stationary process to the

point x. An equivalence relation similar to the one in Denition 4.5 allows us to compare the

empirical distributions obtained from zooming about dierent points, and leads in 6.3 to the

concept of discrete uniform scaling scenery.

A notational remark: we use the symbol θ alternatively for measures that are dened on a cube

of side-length 1 in Rd, and on the entire Rd. When required, it can be assumed that a measure

on a cube is actually dened on the entire space by extending it to be zero outside its cube.

6.1. Cubes and framesets. A discrete scaling scenery is obtained by inating a measure

on a decreasing sequence of cubes. We term such a sequence frameset.

Denition 6.1. A set C ⊂ Rd such that∏di=1 (xi, xi + s) ⊂ C ⊂

∏di=1 [xi, xi + s] for some

x = (x1, . . . xd) ∈ Rd and s > 0, will be called a cube of side-length s.

Denition 6.2. If A and B are cubes in Rd, we denote by H 〈A,B〉 : Rd → Rd the unique

homothety such that H 〈A,B〉∣∣ Ais a bijection between

A and

B.

Denition 6.3. Choose 2 6 p ∈ N. A family Fx = Qx,nn∈N of sets in Rd will be called a

p-frameset decreasing to x ∈ Rd if

(1) Each Qx,n is a cube of side-length p−n

(2) Qx,n+1 ⊂ Qx,n for all n ∈ N(3) x =

⋂n∈N

Qx,n


With each p-frameset Fx = Qx,nn∈N Qn in a p-frameset we associate the sequence of mappings

H 〈Qx,n , Qx,0〉n∈N, where H 〈Qx,n , Qx,0〉 : Rd → Rd is the unique homothety from Qx,n to

Qx,0 described above.

Examples

(1) The CP p-frameset around x ∈ QFix some 2 6 p ∈ N. Recall the denition and notation of the p-ary decomposition

of Q (5.3). For x ∈ Q and n ∈ N, dene QCPx,n = Qwn(x), the unique cube of side

p−n containing x in the p-ary decomposition of Q. Clearly, FCPx =QCPx,n

n∈N is a p-

frameset decreasing to x. Note that QCPx,0 = Q. We call this frameset the CP p-frameset

around x ∈ Q. The grid-size parameter p is implicit in our notation.

(2) The discrete centered frameset around x ∈ Q.Again x 2 6 p ∈ N and x = (x1, . . . xd) ∈ Rd. Dene

Qx,n =d∏i=1

[xi −

12p−n , xi +

12p−n

].

Clearly, Fx =Qx,n

∞n=0

is a p-frameset decreasing to x. We call it the centered p-

frameset around x. This notation is meant to conform with that of Denition 4.1. As

in that denition, we use the simplied symbol Qx for Qx,0 =∏di=1

[xi − 1

2 , xi + 12

].

6.2. Discrete scaling scenery and its empirical distribution.

Denition 6.4. Let Fx = Qx,nn∈N be a p-frameset and suppose that θ ∈M(Rd)is a Radon

measure on Rd, such that θ (Qx,n) > 0 for all n ∈ N. The sequence θx,nn∈N of probability

measures on Qx,0, dened by

θx,n =H 〈Qx,n , Qx,0〉 θ

θ (Qx,0)

∣∣∣Qx,0

(restricted to Qx,0), will be called the Fx-scaling scenery, or simply the Fx-scenery, of θ.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1: The Cubes QCPx,n (blue) and Qx,n (red) for x = (0.39, 0.39) ∈ Q (green) and

n = 1, 2, 3, 4.

Examples

(1) If Fx = Qx,nn∈N is any p-frameset and λ is the Lebesgue measure on Rd, then the

Fx-scenery is dened. It is the constant sequence θn ≡ θ∣∣Q.

(2) CP-scenery of θ ∈ P (Q) around x:

Let (Φ, T, µ) be a CP-system with grid-size parameter p and choose some (θ, x) ∈ Φ.

Let FCPx be the CP p-frameset around x. By the denition of Φ, the FCPx -scenery of θ

is dened. Denote it byθCPx,n

n∈N. The transformation T generates the FCPx -scenery

of θ in the sense that if Tn (θ, x)=(θn, xn) then for all n we have

θn = θCPx,n .

(3) Discrete central scenery of θ ∈M(Rd)around x:


Let θ ∈ M(Rd)and x ∈ Rd. It is easy to verify that the Fx -scenery of θ around x

is dened if and only if x ∈ |θ|. In this case, writeθx,n

n∈N for the Fx -scenery of θ

around x.

Observe that if (Φ, T, µ) is a CP-system and (θ, x) ∈ Φ, then x ∈ |θ|. Therefore, for (θ, x) ∈ Φ

both the FCPx -scenery and the Fx -scenery of θ around x appear to be dened. However note if

(θ, x) ∈ Φ and x is in the interior of Q, the measures θx,n are only dened for large enough n. In

7, we remedy this diculty by considering extended CP-Systems, and nd a transformation

that maps the measure θx,n to the measure θCPx,n .

Denition 6.5. Let Fx = Qx,nn∈N be a p-frameset, and let θ ∈ M(Rd)be a Radon

measure on Rd. Suppose that θnn∈N, the Fx-scenery of θ, is dened. Consider x = θnn∈N

as a signal, taking values in the compact space17 X = P (Qx,0). If the signal x has limiting

empirical means in the sense of Denition 3.3, we say that θ has Fx-limiting empirical means.

The corresponding empirical distribution ηx, which is a probability measure on P (Qx,0)N, is

called the Fx-empirical distribution of θ and denoted η [Fx, θ].

6.3. Uniform discrete scaling scenery. We now dene the discrete analogy to the notion

of uniform continuous-scale scenery from 4.1.

Our notion of continuous-scale scenery used only concentric cubes decreasing to a point. This

associated to each point x a frameset decreasing to x. What about our general discrete setting?

Consider the example of a CP-System. Choose some θ ∈ P (Q). Around any x ∈ Q such that

(θ, x) ∈ Φ, we have the CP-frameset decreasing to x. This is more than a denition of an

individual frameset: rather, we attach to any x ∈ Q with (θ, x) ∈ Φ a frameset decreasing to

x. We may thus investigate the behavior of the empirical distribution µ[FCPx , θ

]as a function

of x. In order to compare empirical distributions over dierent points, it is thus necessary to

explicitly dene the association x 7→ Fx, where Fx is a frameset decreasing to x.

Since we are interested in small-scale structure of measures, we only consider framesets that

are decreasing to points in the cube Q = [0, 1]d.

Denition 6.6. Let θ ∈ M(Rd)be a Radon measure on Rd. A frameset-system F for θ is a

map xF7−→ Fx dened on a subset of Q such that for each x, Fx is a frameset decreasing to x,

and such that for any x in the domain F , the Fx-scenery of θ is dened.

17see 2.1


Examining the foregoing example of the CP-frameset about x ∈ Q, denoted by FCPx (the grid-

size parameter p is understood), we nd that FCP : x 7→ FCPx is in fact a frameset-system. Let

us conveniently call it the CP-frameset-system. Similarly, examining the example of Fx , thecentral frameset about x ∈ Q, we see that F : x 7→ Fx is a frameset-system, which we shall

call the central frameset-system.

For θ ∈ M(Rd)and a frameset-system F for θ, we have thus constructed an additional

structure above θ: above any point x ∈ |θ| ∩Q such that θ has Fx-limiting empirical means,

we place the P (Qx,0)-valued18, discrete-time stationary process η [Fx, θ].

As in the case of continuous-scale scenery, we are interested in the case where the mapping

x 7→ η [Fx, θ] is θ-a.e dened and constant on Q. Again we need a way to compare the

processes η [Fx, θ] and η [Fy, θ] for x 6= y in the domain of F , although they have dierent

state-spaces: one is P (Qx,0) -valued and the other is P (Qy,0) - valued.

Denition 6.7. Let x, y ∈ Rd and let ηx (resp. ηy) be distributions of discrete-time, P (Qx,0)

(resp. P (Qy,0))-valued stationary processes19. Suppose that whenever the process θn is dis-

tributed ηx, the process H 〈Qx,0 , Qy,0〉 θn is distributed ηy20. We then say that ηx and ηy

are equivalent and write ηx ∼ ηy.

The relation ∼ is easily seen to be an equivalence relation. Let us write η∼ for the equivalence

class of a distribution η. As in the continuous case we have

Lemma 6.8. Let θ ∈ M(Rd). Assume that F is a frameset-system for θ and that x, y are in

the domain of F . Let (θx,n)n∈N and (θy,n)n∈N be the Fx and Fy sceneries of θ around x and

y. If for some cube Q′ of side-length 1 the P (Q′)-valued signals (H 〈Qx,0 , Q′〉 θx,n)n∈N and

(H 〈Qy,0 , Q′〉 θy,n)n∈N both have limiting empirical means with the same empirical distribu-

tion, then:

(1) θ has Fx and Fy limiting empirical means, and

(2) η [Fx, θ] ∼ η [Fy, θ] .

18Here Fx = Qx,nn∈N19In other words, ηx is a shift-invariant probability measure on P (Qx,0)N

20This amounts to the statement that H 〈Qx,0 , Qy,0〉 induces an isomorphism of the systems (P (Qx,0) , ηx, σ)

and (P (Qy,0) , ηy, σ) where σ is the shift.


Now that we are able to compare the empirical distribution of θ around dierent points, we

arrive at the central denition of this section. Compare the following to Denition 4.7:

Denition 6.9. Let θ ∈ M(Rd)and let F be a frameset-system for θ dened θ-a.e. inside

Q = [0, 1]d. We say that θ has uniform F-scenery if for θ-almost every x ∈ Q, θ has Fx-empirical means, and the mapping x 7→ η [Fx, θ]∼ is θ-a.e. constant on Q. In this case, any

representative of the equivalence class that is the (almost everywhere) constant value of this

mapping is called the F-uniform empirical distribution, and denoted by η [F , θ].

Note that if θ has uniform F-scenery, then the uniform empirical distribution η [F , θ] can be

chosen at convenience to be a probability measure on P (Q0)N or P (Q)N.

As in the continuous case (Lemma 4.8), we can combine Lemma 3.5 and Lemma 6.8 to obtain

a practical criterion for the discrete uniform scaling scenery property:

Lemma 6.10. Let θ ∈ M(Rd)and let F be a frameset-system for θ dened θ-a.e. inside

Q = [0, 1]d. Write Qx,nn∈N for the frameset Fx and θx,nn∈N for the Fx-scenery of θ. Let

Q′ be a cube of side-length 1 (call it the test cube) and assume that A ⊂ C (P (Q′)) is a dense

subalgebra with 1Q′ ∈ A. If there exists a set H ⊂ Q with θ (Q \H) = 0 such that for any

` ∈ N and any continuous functions f0, . . . , f` ∈ A, the limit

limN→∞

1N

N−1∑n=0

[∏i=0

fi(H⟨Qx,0 , Q

′⟩ θx,n+i

)]exists and is identical for each x ∈ H, then θ has uniform F- scenery and the uniform empirical

distribution η = η [F , θ] is uniquely determined by the values

ˆ

P(Q′)R

[∏i=0

fi (yi)

]dη(y) .

6.4. Generic points of a CP-System and uniform CP-scaling scenery. As we have seen

in 3.3, generic points in a measure-preserving system are points whose orbits are signals with

limiting empirical means. We proved that the set of generic points is always of full measure,

and that in an ergodic system, the empirical distribution of any orbit starting in a generic point

is identical.

We now use a variation on this theme to show that almost surely, measures in an ergodic

CP-System have uniform FCP -scaling scenery.


Let (Φ, T, µ) be a CP-System (the grid-size parameter p is understood). By Corollary 2.6,

we can nd a countable, dense subalgebra A = gi∞i=1 of C (P(Q)). For any ` ∈ N and any

`+ 1-tuple k = (k0, . . . , k`) ∈ N`+1, dene gk : Φ→ R by

gk =∏i=0

gki π Ti

where π : Φ → P(Q) is the projection, given by (θ, x) π7→ θ. Clearly gk ∈ L1 (Φ, µ). By

Birkho's ergodic theorem, there exists a set Gk ⊂ Φ with µGk = 1 such that for all (θ, x) ∈ Gk,the limit

(6.1) limN→∞

1N

N−1∑n=0

[gk Tn (θ, x)] =ˆ

Φ

gk dµ

exists. Dene

G =⋂`∈N

⋂k=(k0,...,k`)∈N`+1

Gk

so that µG = 1. Suppose that µ is adapted to ν ∈ P (P (Q)). By 5.3, there exists a set

T ⊂ P (Q) with νT = 1 such that for each θ ∈ T , and for θ-almost every x ∈ Q, we have

(θ, x) ∈ G.

The set G takes the place of the generic set (in the sense of 3.3) of the CP-System (Φ, T, µ).

We call elements of set T the generic measures.

Collecting several of our results so far, we now prove that a generic measure in an ergodic

CP-System has uniform FCP -scaling scenery. In fact, we prove a much stronger result: not

only is the empirical distribution η[FCPx , θ

]constant for θ-a.e x ∈ Q (for xed θ), but if fact

it is identical for all generic measures.

Theorem 6.11. Let (Φ, T, µ) be an ergodic CP-System and let G ⊂ Φ be its generic set.

Then for every (θ, x) ∈ G, θ has FCPx limiting empirical means, and the mapping (θ, x) 7→µ[FCPx , θ

]∼ is constant on G. In particular, every generic measure θ ∈ P (Q) has uniform

FCP -scenery.

Proof. We use the notation G and A = gi∞i=1 introduced above. Choose some generic measure

θ ∈ P (Q). Then for θ-a.e. x ∈ Q, (θ, x) is a generic point. Observe that we always have

gk Tn (θ, x) =∏i=0

gk π Tn+i (θ, x) =∏i=0

gki(θCPx,n+i

),


hence if (θ, x) is a generic point, then by (6.1) we have for ` ∈ N and any ` + 1-tuple k =

(k0, . . . , k`) ∈ N`+1

limN→∞

1N

N−1∑n=0

[∏i=0

gki(θCPx,n+i

)]= lim

N→∞

1N

N−1∑n=0

[gk Tn (θ, x)] =

=ˆ

Φ

gk dµ .

The right-hand side depends neither on θ nor on x, and therefore so does the limit on the left.

Thus, if θ is a generic measure then we are in the conditions of Lemma 6.10 (with Q as test

cube), and deduce that θ has uniform FCP -scenery. Denote its uniform empirical distribution

by η[FCP , θ

]. By denition,

limN→∞

1N

N−1∑n=0

[∏i=0

gki(θCPx,n+i

)]=ˆ

P(Q)

[∏i=0

gki (yi)

]dη[FCP , θ

](y)

but this implies that

ˆ

P(Q)

[∏i=0

gki (yi)

]dη[FCP , θ

](y) =

ˆ

Φ

gk dµ

which does not depend on the choice of our generic measure θ. Since, by Lemma 6.10, the

uniform empirical distribution η[FCP , θ

]is determined by all valued

ˆ

P(Q)

[∏i=0

gki (yi)

]dη[FCP , θ

](y)

for each ` ∈ N and k ∈ N`+1, this implies that the uniform empirical distribution η[FCP , θ

]is

independent of the choice of the generic measure θ. This amounts to the statement that the

map (θ, x) 7→ η[FCPx , θ

]∼ is constant on G.

Remark 6.12. As indicated in 5.1, CP-Systems are equivalent to Markov processes on the state

space P (Q). As such, the ergodic decomposition of a non-ergodic CP-System (Φ, µ, T ) can

be understood using the ergodic decomposition of Markov processes (for example [D, pp.214]).

In particular, it can be shown that for a given θ ∈ P (Q), almost all (θ, x) ∈ Φ must belong

to the same ergodic component of Φ. In other words, the dierent measures θ are divided

among the ergodic components (indeed, the ergodic components of (Φ, µ, T ) are determined by

transition operator of the underlying Markov process). Using Theorem 6.11 for each ergodic

component of (Φ, µ, T ) as an ergodic CP-System in turn, we then nd that ν-almost every21

21µ is adapted to the measure ν on P (Q).


θ ∈ P (Q) has uniform FCP -scenery, η[FCP , θ

]. Thus, even if a CP-System is not ergodic, a

generic measure still has uniform FCP -scenery. However, the map (θ, x) 7→ η[FCPx , θ

]will be

(almost everywhere) constant on each ergodic components, instead of being constant (almost

everywhere) on Φ.

7. Centered scaling scenery in extended CP-systems

7.1. Extended CP-Systems. We use the notation of 5.3. Fix a grid-size parameter 2 6 p ∈ Nand consider the set

Φ =

(θ, x)∣∣ θ (Q) = 1 ; ∀n. θ (Qn(x)) > 0

⊂M

(Rd)×Q ,

equipped with the subspace topology induced by the product topology onM(Rd)×Q.

Dene a transformation T : Φ→ Φ by

(7.1) T (θ, x) =(ψλ(x) (θ)θ (Q1(x))

, px− λ(x))

where ψλ and λ(x) are as dened in 5.3.

In analogy to the denition of a CP-System, we say that a measure µ onM(Rd)×Q is adapted

if there is a measure ν onM(Rd)such that dµ(θ, x) = dθ(x) dν(θ), i.e

(7.2)

ˆ

Φ

f(θ, x) dµ(θ, x) =ˆ

M(Rd)

ˆQ

f(θ, x) dθ(x)

dν(θ) .

Denition 7.1. Choose a grid-size parameter p and let Φ and T be as above. Let µ be a measure

on Φ. The system (Φ, T , µ) is called an Extended CP-System if µ is an adapted, T -invariant

measure on Φ.

In analogy to Proposition 5.3, we have

Proposition 7.2. Suppose that(Φ, T , µ

)is an extended CP-system, and µ is adapted to ν. If

an event occurs for µ-almost every (θ, x) ∈ Φ, then it occurs for ν-almost every θ ∈ M(Rd),

and for every such θ, it occurs θ-almost every x ∈ Q.


We now observe that an extended CP-System (Φ, T , µ) contains as a factor a CP-System of the

same grid-size parameter p.

Denition 7.3. The map π : Φ→ Φ dened by

(7.3) π : (θ, x) 7→(θ∣∣Q, x)

is called the restriction map.

Note that we always have π T = T π. Suppose that (Φ, T , µ) is an extended CP-System

and set µ = π µ, where π : Φ → Φ is the restriction map. Then µ is a measure on Φ, and

a routine check shows that it is T -invariant, namely that (Φ, T, µ) is a CP-System. It follows

that (Φ, T, µ) is a factor of (Φ, T , µ), with respect to the factor map π :(Φ, T , µ

)→ (Φ, T, µ).

Now, if (Φ, T , µ) is an ergodic extended CP-System, then the above argument shows that there

exists an ergodic CP-System (Φ, T, µ), for which the restriction map π :(Φ, T , µ

)→ (Φ, T, µ)

is a factor map. Interestingly, for ergodic CP-Systems, this correspondence can be reversed:

an ergodic CP-System has an extension in the form of an ergodic extended CP-System, where

the factor map is the restriction π :(Φ, T , µ

)→ (Φ, T, µ).

Recall from 5.5 that a CP-System (Φ, T, µ) is called non-degenerate if for µ-a.e. (θ, x) ∈ Φ,

θ (∂Q) = 0, and that an ergodic CP-System can always be assumed to be non-degenerate.

Theorem 7.4. Let (Φ, T, µ) be a non-degenerate, ergodic CP-System with grid-size parameter

p. Then there exists an ergodic extended CP-System (Φ, T , µ) with the same grid-size parameter

such that µ = πµ, where π : Φ→ Φ is the restriction map, and in fact π : (Φ, T , µ)→ (Φ, T, µ)

is a factor map.

The proof is technical and deferred to the appendix. The basic idea is to use the Rohlin

natural extension22 of (Φ, T, µ) in order to construct the extended system (Φ, T , µ). Note that

the extended system (Φ, T , µ) is not invertible; we do not need it to be. We also remark that

the extension (Φ, T , µ) in Theorem 7.4 is essentially unique.

22see 2.4


7.2. Discrete central scenery in CP-Systems. As indicated in 6.2, since measures in

extended CP-Systems are dened on the entire Rd (rather than on Q) we can consider the

Fx -scenery of such measures around any point x ∈ Q in their support, even for points x

in the vicinity of ∂Q. If (Φ, T , µ) is an extended CP-System and (θ, x) ∈ Φ, then both the

FCPx -scenery and the Fx -scenery of θ are dened.

Measures on Rd may be translated freely, so it becomes easy to relate the the Fx -scenery to

the FCPx -scenery around any point x ∈ Q. Below, we nd the measure θx,n in the Fx -sceneryof θ around x as a function of the measure θCPx,n from the FCPx -scenery of θ. We then use

this fact to prove the analog of Theorem 6.11 for central scenery, namely that the mapping

(θ, x) 7→ η[Fx , θ

]∼ is a.e. constant in an ergodic extended CP-system.

Here is a review of the notational zoo involved. Let (Φ, T, µ) be an ergodic CP-system (the

grid-size parameter p is understood) and let(Φ, T , µ

)be the ergodic extended CP-System from

Theorem 7.4. Fix some (θ, x) ∈ Φ. We denote the central p-frameset of θ decreasing to x by

Fx =Qx,n

n∈N, where

Qx,n =d∏i=1

[xi −

12p−n , xi +

12p−n

],

and byθx,n

n∈N the Fx -scenery of θ. We abbreviate Qx,0 by Qx. Similarly, denote by

FCPx =QCPx,n

n∈N the CP-frameset decreasing to x , and by

θCPx,n

n∈N the FCPx -scenery of θ.

The following lemma is easy to verify:

Lemma 7.5. If(θCPx,n , xn

)= Tn (θ, x) then we have

(1) H⟨Qx , Q

x,n

⟩(y) = p−ny + (1− p−n)x

(2) H⟨Q , QCPx,n

⟩(y) = p−n (y − xn) + x.

We now relate the measure θx,n to the measure θCPx,n . Obviously, they are both blow-ups by pn

of the same measure θ. They dier by translation of the origin, and by the cube on which they

are normalized.

Lemma 7.6. For any measurable A ⊂ Rd,

θx,n (A) =θCPx,n (A− x+ xn)

θCPx,n (Qxn)


Proof. By Lemma 7.5

H⟨Qx, Q

x,n

⟩(A) = p−nA+ (1− p−n)x =

= p−n (A− xn + xn − x) + x = H⟨Q , QCPx,n

⟩(A+ xn − x) ,

so up to normalization we have

θx,n (A) ∝ θ(H⟨Qx, Q

x,n

⟩A)

= θ(H⟨Q , QCPx,n

⟩(A− x+ xn)

)∝ θCPx,n (A− x+ xn) .

But sinceθCPx,n (Qx − x+ xn)

θCPx,n (Qxn)= 1 ,

the above normalization is correct, and we are done.

Recall that M(Rd)is the set of Radon measures on Rd equipped with the weak-* topology

that is induced by Cc(Rd), the space of compactly supported, continuous functions on Rd.

Denition 7.7. For u ∈ Rd, dene τu :M(Rd)→M

(Rd)by

(7.4)

ˆ

Rd

f(x) d (τuθ) (x) =ˆ

Rd

f(x− u) dθ(x) for f ∈ Cc(Rd).

This is just the translation of a measure by u.

Denition 7.8. Suppose that(Φ, T , µ

)is an extended CP-system . For A ⊂ Q× Φ given by

A =

(y , (θ, x))∣∣ y ∈ |θ|, dene ψ : A→M

(Rd)by

(7.5) ψy (θ, x) =1

θ (Qy)τ(x−y)θ

∣∣∣Qy

restricted to a measure on Qy.

To simplify notation, we write ψy (θ, x) instead of ψ (y , (θ, x)). Basically, ψ translates θ by y,

restricts and normalizes it to obtain a probability measure on the unit cube centered in y.

The next lemma shows that mapping ψ is crucial to our discussion, as it maps the CP-scenery

to the central scenery:

Lemma 7.9. Let(Φ, T , µ

)be an extended CP-System. Suppose that (θ, x) ∈ Φ is such that the

Fx -scenery of θ is dened. Then for all n ∈ N we have


(1) θx,n = ψx(θCPx,n , xn

)= ψx Tn (θ, x) and

(2) H 〈Qx , Q0〉 θx,n = ψ0 Tn (θ, x)

Proof. (1) is just a restatement of Lemma 7.623 (2) we have

H 〈Qx , Q0〉 θx,n = τx

(θx,n

)= τx ψx Tn (θ, x) ,

where τx is dened in (7.4). However, clearly we have τx ψx = ψ0.

Recall from 2.1 that the mapping θ 7→ θ (Qy), for some xed y, is not necessarily weak-

* continuous as a function of θ. Thus, unfortunately, the mapping ψy (·) is not in general

continuous. (This may be avoided if, instead of normalizing by the indicator function of Q0 we

would normalize using some smooth version of it; we do not treat this alternative denition).

It is therefore important to verify that ψ is, at least, measurable. The next two lemmas show

that ψ is a pointwise limit of continuous function, and hence measurable.

Lemma 7.10. The mapping τ : Q×M(Rd)→M

(Rd), τ : (y, θ) 7→ τy (θ) dened in (7.4) is

continuous in both y and θ.

Proof. Fix θ ∈ M(Rd)and let us show continuity at an arbitrary point y ∈ Q. Let V be a

basis neighborhood of τy (θ). It is enough to nd a neighborhood U of y such that τU (θ) ⊂ V .Now,

V =ν∣∣ ∣∣∣∣ˆ fidν −

ˆfidτy (θ)

∣∣∣∣ < ε , i = 1 . . . N

for some ε > 0, N and f1, . . . fN ∈ Cc(Rd). Let K ⊂ Rd be a compact set large enough to con-

tain the supports of f1, . . . fN . Since each fi is continuous on K, it is uniformly continuous, so

we can choose δ > 0 such that ‖x− y‖ < δ =⇒ |fi (x)− fi (y)| < εθ(K) for all i = 1, . . . , N . Let

us show that the neighborhood U = (y − δ, y + δ) satises τU (θ) ⊂ V . Indeed, if |y − z| < δ,

then for all i = 1, . . . , N∣∣∣∣ˆ fi(x)dτy (θ) (x)−ˆfi(x)dτz (θ) (x)

∣∣∣∣ =∣∣∣∣ˆ fi(x− y)dθ(x)−

ˆfi(x− z)dθ(x)

∣∣∣∣ 66ˆ|fi(x− y)− fi(x− z)| dθ(x) < ε

which means that τz (θ) ∈ V . Thus y 7→ τy (θ) is continuous for xed θ. A similar argument

shows that θ 7→ τy (θ) is continuous for xed y.

23Since x ∈ |θ| for each (θ, x) ∈ Φ, the map ψx is dened.


Lemma 7.11. Let f : Rd → R be a continuous, compactly supported function, and suppose

that f∣∣Q0≡ 1. Dene A ⊂ Q × Φ by A =

(y, (θ, x))

∣∣ θ (Qy) > 0. Then the mapping

ψ[f ] : A→M(Rd)dened by

(7.6) ψ[f ]y (θ, x) =1´

Rd f (x− y) dθ(x)τ(x−y)θ .

is continuous.

Proof. Since ˆ

Rd

f (x− y) dθ(x) =ˆ

Rd

f (x) dτyθ(x) ,

the denominator is a composition of the mapping (y, (θ, x)) 7→ τyθ (continuous by the previous

lemma) and the mapping θ 7→´

Rd f dθ (weak-* continuous by denition). Hence the denomina-

tor is given by a continuous function whenever it is nonzero. The mapping (y, (θ, x)) 7→ τ(x−y)θ

is continuous by the previous lemma.

Combining the last two results, we get the measurability of ψ:

Lemma 7.12. ψ : A→M(Rd), where A =

(y, (θ, x))

∣∣ y ∈ |θ|, is Borel measurable.

Proof. Let hm : Rd → R be a series of continuous, compactly supported functions converg-

ing pointwise to 1Q0 , such that for all m we have 0 6 1Q0 6 gm 6 1. (This is a standard

construction; see for example [Bil, Theorem 1.2]). We now nd a sequence (gm)∞m=1 of mea-

surable functions, gm : A → M(Rd), which converges to pointwise to ψ. Indeed, dene

gm : (y, (θ, x)) 7→ ψ[hm]y (θ, x). By the above lemma, for each m, gm is continuous and hence

Borel measurable. Since hm (x− y)→ 1Qy(x) pointwise, by the bounded convergence theorem

we have

limm→∞

ˆ

Rd

hm (x− y) dθ(x) =ˆ

Rd

1Qy(x) dθ(x) = θ (Qy)

so that for all (y, (θ, x)) ∈ A,

limm→∞

ψ[gm]y (θ, x) = limm→∞

1´Rd f (x− y) dθ(x)

τ(x−y)θ = ψy (θ, x)

and it follows that ψ is a pointwise limit of a sequence of measurable functions, hence measurable

itself.


We have thus established that the mapping ψ obeys the laws of measure theory. By Lemma

7.9, ψ becomes the bridge between the CP-scenery and the central scenery. We now use

this bridge to prove an analog of Theorem 6.11 for central scenery, which stipulates that the

mapping (θ, x) 7→ η[Fx , θ

]is a.e. dened and constant on Q in an ergodic extended CP-

system.While this theorem will not be used explicitly in the sequel, it morally precedes our

main result, Theorem 8.1, and contains some of the main ideas in the proof of the latter.

Theorem 7.13. Let(Φ, T , µ

)an ergodic extended CP-system (the grid-size parameter p is

understood). Then there exists a set G ⊂ Φ with µG = 1 such that for every (θ, x) ∈ G, θ has

Fx limiting empirical means and such that the mapping (θ, x) 7→ η[Fx , θ


In particular, for every (θ, x) ∈ G, we nd that θ has uniform Fx -scenery.

Proof. By Corollary 2.6, we can nd be a countable, dense subalgebra A of the separable space

C (P (Q0)). For any ` ∈ N and any `+ 1-tuple k = (k0, . . . , k`) ∈ N`+1, dene gk : Φ→ R by

(7.7) gk (θ, x) =∏i=0

gki ψ0 T i (θ, x) .

where ψ0 is dened in (7.5). By Lemma 7.12, gk is measurable. As it is clearly bounded, it

follows that gk ∈ L1(Φ, µ

). The extended CP-System

(Φ, T , µ

)is ergodic by assumption, so we

can invoke the Birkho ergodic theorem to obtain a set Gk ⊂ Φ such that for any (θ, x) ∈ Gk,

limN→∞

1N

N−1∑n=0

gk Tn (θ, x) =ˆ

Φ

gkdµ .

Dene

G =⋂`∈N

⋂k∈N`+1

Gk

so G ⊂ Φ and µG = 1. Now, choose some (θ, x) ∈ G, and let us show that θ has Fx -empirical

means. By Lemma 7.9,

H 〈Qx , Q0〉 θx,n = ψ0 Tn (θ, x) .

For ` ∈ N and k = (k0, . . . , k`) ∈ N`+1 we then have

∏i=0

gki

(H 〈Qx , Q0〉 θx,n+i

)=

∏i=0

gki ψ0 Tn+i (θ, x) =

= gk Tn (θ, x)


where g is given by (7.7). We nd that any(θ, x) ∈ G has the property that for any ` ∈ N and

k ∈ N`+1,

limN→∞

1N

N−1∑n=0

∏i=0

gki

(H 〈Qx , Q0〉 θx,n+i

)= lim

N→∞

1N

N−1∑n=0

gk Tn (θ, x)

=ˆ

Φ

gkdµ ,

and the limit on the left is independent of (θ, x). Since µG = 1, by Proposition 7.2 we can

assume that if (θ, x) ∈ G, then also (θ, y) ∈ Φ for θ-a.e. y ∈ Q. Fix some (θ, x) ∈ G. Since forθ-a.e. y ∈ Q we have (θ, y) ∈ G, we nd that for any ` ∈ N and any k ∈ N`+1, the limit

limN→∞

1N

N−1∑n=0

∏i=0

gki

(H 〈Qy , Q0〉 θy,n+i

)exists and is identical for θ-almost every y ∈ Q. We now invoke Lemma 6.10 (with Q0 as

test cube), and obtain that θ has F-uniform scaling scenery, and that the uniform empirical

distribution η[F, θ


ˆ

P(Qx)N

∏i=0

gki (y) η[Fx , θ

](y) .

But by the denition of η[Fx , θ

],

limN→∞

1N

N−1∑n=0

∏i=0

gki

(θx,n+i

)=ˆ

P(Qx)N

∏i=0

gki (y) η[Fx , θ

](y)

which turned out to be independent of the choice of (θ, x) ∈ G. We deduce that the mapping

(θ, x) 7→ η[Fx , θ


8. Uniform continuous-scale scenery in ergodic CP-Systems

We nally connect the uniform continuous-scale scenery property, which was introduced in 4,

to the (seemingly unrelated) notion of CP-Systems, which was developed in 5. The main

result states that almost any (extension of a) measure in a non-degenerate ergodic CP-System

has uniform continuous-scale scenery.


This is achieved as follows. We use Theorem 7.4 to extend a non-degenerate ergodic CP-

System to its corresponding extended ergodic CP-System. We follow the proof of Theorem

7.13 to make the transition from the CP-scenery to the central scenery. Finally, we note that

the continuous-scale scenery is a function of the discrete central scenery. Roughly speaking,

this means that the continuous-scale scenery, or FRx -scenery, of θ (for (θ, x) ∈ Φ) is given by

a stationary transformation of the FCPx -scenery of θ. The result then follows from ergodicity.

Recall the notation of 4.1 for the continuous-scale scenery of θ around x ∈ |θ|.

Theorem 8.1. Let (Φ, T, µ) be a non-degenerate ergodic CP-System (p is understood), and let(Φ, T , µ

)be the corresponding ergodic extended CP-System. Then there exists a set G ⊂ Φ with

µG = 1 such that for every (θ, x) ∈ G, the measure θ has FRx limiting empirical means, and

the mapping (θ, x) 7→ η[FRx , θ

]∼ is constant on G. In particular, for every (θ, x) ∈ G, we nd

that θ has uniform FRx -scenery.

Proof. First note that if(Φ, T , µ

)is an extended CP-System and (θ, x) ∈ Φ, then x ∈ |θ|, so

the FRx -scenery (the continuous-scale scenery) of θ around x is dened.

By Corollary 2.6 we can nd a countable, dense subalgebra A in C (P (Q0)). For ` ∈ N,g = (g0, . . . , g`) ∈ A and τ = (τ0, . . . , τ`) ∈ (Q)`+1, dene Ψg;τ : Φ→ R by

(8.1) Ψg;τ (θ, x) =

t=1ˆ

t=0

∏i=0

gi

([ψ0(θ, x)]0,t+τi

)dt .

The following proposition shows that Ψg;τ is well dened, measurable (indeed integrable), and

most importantly that the ergodic averages of Ψg;τ are exactly the limits we need to calculate.

Proposition 8.2. For ` ∈ N, g = (g0, . . . , g`) ∈ A and τ = (τ0, . . . , τ`) ∈ (Q)`+1, the function

Ψg;τ : Φ→ R satises the following:

(1) The integrand is dened for all (θ, x) ∈ Φ and integral in (8.1) exists

(2) Ψg;τ is Borel measurable and in fact, Ψg;τ ∈ L1(Φ, µ

)(3) For all N ∈ N we have

(8.2)1N

N−1∑n=0

Ψg;τ Tn (θ, x) =1N

t=Nˆ

t=0

∏i=0

gi

[H 〈Qx , Q0〉 θx,t+τi

]dt ,


(4) If the limit 1N

∑N−1n=0 Ψg;τ Tn (θ, x) exists, so does the limit

limT→∞

1T

t=Tˆ

t=0

∏i=0

gi


]dt

and the two are equal.

Let us show how the rest of the proof follows from this proposition. Let(Φ, T , µ

)be the ergodic

extended CP-System corresponding to the given system (Φ, T, µ). Fix `, g ∈ A and τ ∈ (Q)`+1.

By the above proposition, Ψg;τ is integrable on(Φ, µ

). Appealing to Birkho's ergodic theorem,

and taking the intersection over all `, g and τ as in the proof of Theorem 7.13, we can nd

G ⊂ Φ with µG = 1 such that for every (θ, x) ∈ G, every ` ∈ N, and g = (g0, . . . , g`) ∈ A and

every rational `+ 1-tuple λ = (λ0, . . . , λ`) ∈ (Q)`+1, the limit

limN→∞

1N

N−1∑n=0

Ψg;τ Tn (θ, x)

exists, and in fact

(8.3) limN→∞

1N

N−1∑n=0

Ψg;τ Tn (θ, x) =ˆ

Φ

Ψg;λ dµ .

Now, Proposition 8.2 (to be proved below) implies that for every (θ, x) ∈ G, every g =

(g0, . . . , g`) ∈ A and every rational `+ 1-tuple λ = (λ0, . . . , λ`) ∈ (Q)`+1, we have

(8.4) limT→∞

1T

t=Tˆ

t=0

∏i=0

gi


]dt =

ˆ

Φ

Ψg;λ dµ ,

and therefore the limit on the left is independent of the choice of (θ, x). Since µG = 1, by

Proposition 7.2 we can assume that if (θ, x) ∈ G, then also (θ, y) ∈ Φ for θ-a.e. y ∈ Q. Fix

some (θ, x) ∈ G. Since for θ-a.e. y ∈ Q we have (θ, y) ∈ G, we nd that for any ` ∈ N, g ∈ Aand λ ∈ (Q)`+1, the limit

limT→∞

1T

t=Tˆ

t=0

∏i=0

gi

[H 〈Qy , Q0〉 θy,t+τi

]dt

exists and is identical for θ-almost every y ∈ Q. We thus nd ourselves in the conditions of

Lemma 4.8, which implies that θ has uniform continuous-scale scenery, and that the uniform


empirical distribution η[FR, θ


ˆ

P(Q0)R

[∏i=0

fi (yτi)

]dη(y) .

But by the denition of η[FR, θ

],

limT→∞

1T

t=Tˆ

t=0

∏i=0

gi


]dt =

ˆ

P(Q0)R

[∏i=0

fi (yτi)

]dη(y) ,

which we found out in (8.4) to be independent of the choice of (θ, x) ∈ G. This means that the

map (θ, x) 7→ η[FR, θ


We still have to prove Proposition 8.2. Toward this end, we observe a few useful properties of

the continuous-scale scenery.

Lemma 8.3. (The semigroup property of the continuous central scenery): For θ ∈ M(Rd)

x ∈ Rd with x ∈ |θ|, we have

θx,s+t =(θx,s

)x,t

.

Proof. Indeed, for any measurable A ⊂ Rd,

θx,s+t (A) =1

θ(Qx,t+s

) (H⟨Qx,t+s , Qx⟩ θ) (A) =

=1

θ(Qx,t+s

)θ (H⟨Qx , Qx,t+s⟩ (A))

=

=1

θ(Qx,t+s

) θ (H⟨Qx,t , Qx,t+s⟩ H⟨Qx , Qx,t⟩ (A))

=

=1

θ(Qx,t+s

) θ (H⟨Qx , Qx,s⟩ H⟨Qx , Qx,t⟩ (A))

=

=1

θ(Qx,t+s

) (H⟨Qx,s , Qx⟩ θ)(H⟨Qx , Qx,t⟩ (A))

=

=θ(Qx,s

)θ(Qx,t+s

) θx,s (H⟨Qx , Qx,t⟩ (A))

=

=1

θx,s(Qx,t

)θx,s (H⟨Qx , Qx,t⟩ (A))

=(θx,s

)x,t

(A)


Lemma 8.4. Let(Φ, T , µ

)be an extended CP-System. Suppose that (θ, x) ∈ Φ with x ∈ |θ|.

Then(ψ0 Tn (θ, x)

)0,t

= H 〈Qx , Q0〉 θx,n+t

Proof. By Lemma 7.9,

(ψ0 Tn (θ, x)

)0,t

=(H 〈Qx , Q0〉 θx,n

)0,t.

It is easy to verify that we always have H 〈Qx , Q0〉 θx,t = (H 〈Qx , Q0〉 θ)0,t. Hence, here

we have (H 〈Qx , Q0〉 θx,n

)0,t

= H 〈Qx , Q0〉 (θx,n

)0,t.

Finally, using Lemma 8.3 we get

H 〈Qx , Q0〉 (θx,n

)0,t

= H 〈Qx , Q0〉 θx,n+t .

Lemma 8.5. Let θ ∈ M(Rd)and 0 ∈ |θ|. The mapping R ×M

(Rd)→ M

(Rd)dened by

(t, θ) 7−→ θ0,t is measurable both in t and in θ.

Proof. As in the proof of Lemma 7.12, this mapping is readily seen to be a pointwise limit of

continuous functions.

We can nally give the

Proof. of Proposition 8.2:

By Lemma 8.5 and Lemma 7.12, the integrand is a composition of measurable and continuous

functions, hence measurable as a function of t. Since it is also bounded by∏`i=0 (sup gi), we

nd that the integral above exists and is nite. A similar argument shows that Ψg;τ itself, as

a function of θ, is measurable. Since it is also bounded, in fact Ψg;τ ∈ L1(Φ, µ

).


Next, for n ∈ N we have

Ψg;τ Tn (θ, x) =

t=1ˆ

t=0

∏i=0

gi

([ψ0 Tn(θ, x)

]0,t+τi

)dt =

=

t=1ˆ

t=0

∏i=0

gi

(H 〈Qx , Q0〉 θx,n+t+τi

)dt =

=

t=n+1ˆ

t=n

∏i=0

gi


)dt ,

where we have used Lemma 8.4, whereby

(ψ0 Tn (θ, x)

)0,t

= H 〈Qx , Q0〉 θx,n+t .

This yields

1N

n−1∑n=0

Ψg;τ Tn (θ, x) =1N

t=Nˆ

t=0

∏i=0

gi


)dt ,

so (8.2) is justied.

Next, assume that the limit

limN→∞

1N

n−1∑n=0

Ψg;τ Tn (θ, x)

exists. We have

limN→∞

1N

n−1∑n=0

Ψg;τ Tn (θ, x) = limN→∞

1N

t=Nˆ

t=0

∏i=0

gi


)dt .

Now, denote M = max06i6` sup |gi|, so

limT→∞

∣∣∣∣∣∣∣1T

dT eˆ

T

∏i=0

gi


)dt

∣∣∣∣∣∣∣ 6 limT→∞

1T

dT eˆ

T

M `+1 = 0

where dT e is the upper integer value of T . But as

1T

T

0

∏i=0

gi (. . .) dt =dT eT· 1dT e

dT eˆ

0

∏i=0

gi (. . .) dt− 1T

dT eˆ

T

∏i=0

gi (. . .) dt,


we obtain from (8.2) that

limT→∞

1T

T

0

∏i=0

gi


)dt =

= limT→∞

dT eT· 1dT e

dT eˆ

0

∏i=0

gi


)dt =

= limN→∞

1N

t=Nˆ

t=0

∏i=0

gi


)dt =

= limN→∞

1N

n−1∑n=0

Ψg;τ Tn (θ, x) .

So we see that if the limit limN→∞1N

∑n−1n=0 Ψg;τ Tn (θ, x) exists, then it must equal

limT→∞

1T

T

0

∏i=0

gi


)dt .

.

Corollary 8.6. The classical Cantor measure has uniform continuous-scale scenery.

Let ν denote the classical Cantor measure on [0, 1]. Note that as in the case of a typical

Brownian occupation measure (4.3), the map x 7→ η[FRx , ν

]∼ is ν-a.e. constant, but not

everywhere constant: if x is an endpoint of a ternary interval, then the empirical distribution

of the FRx -scenery diers from the uniform empirical distribution.

9. Self-similarity and uniform scaling scenery

In the context of fractal geometry, the notion of self-similarity is given dierent formal inter-

pretations. For example, attractors of certain iterated function systems demonstrate 'exact'

self-similarity, while sample paths of certain stochastic processes demonstrate 'statistical' self-

similarity.

The property of uniform continuous-scale scenery appears to be related in some sense to the

notion of self-similarity. This connection is made precise by two observations. The rst ob-

servation, already made in 4.3, is that a typical Brownian occupation measure (a prototype


for 'statistical' self-similarity) has uniform continuous-scale scenery. The second, which we de-

velop here, is that the measures associated to attractors of ane iterated function systems (a

prototype for 'exact' self-similarity) also have uniform scaling scenery.

These observations are obviously too narrow to be conclusive. Nevertheless, they may be

understood as partial evidence that the property of uniform scaling scenery may be used to

unify dierent formal interpretations of the term self-similarity.

We thus aim to prove

Theorem. Let φ1, . . . , φn : Q → Q be an iterated function system of contracting homotheties

(Q ⊂ Rd is the unit cube and n > 2), satisfying the open set condition, with attractor

A =n⋃i=1

φi (A) .

Let θ be a self-similar probability measure on A corresponding to the strictly positive probability

vector (p1, . . . , pn). Then θ has uniform continuous-scale scenery.

The proof is an application of Theorem 8.1 and an excellent example of the usefulness of

the CP-System concept. We rst give the necessary denitions and state without proof basic

results.

9.1. Iterated Function Systems.

(1) Let X ⊂ Rd be a closed set. A map φ : X → X is called a contraction if

‖φ(x)− φ(y)‖ < c ‖x− y‖

for some 0 < c < 1, and a contracting ane map24 with contracting factor 0 < c < 1 if

‖φ(x)− φ(y)‖ = c ‖x− y‖ .

(a) A family of contractions φ1, . . . , φn : X → X is called an Iterated Function System

(IFS). If φ1, . . . , φn are all contracting ane maps, it is called a contracting ane

IFS.

(b) A compact set A ⊂ X is called an attractor of the IFS (φ1, . . . , φn) if

A =n⊎i=1

φi (A)

24Sometimes called contracting homothety, or similitude.


(c) A measure µ on an attractor A of the IFS (φ1, . . . , φn) is called a self-similar

measure associated to the probability vector (p1, . . . , pn) if

µ =n∑i=1

pi · φi µ

(d) The IFS (φ1, . . . , φn) is said to satisfy the open set condition if there exists V ⊂ Xopen, such that

⋃ni=1 φi (V ) ⊂ V and φi (V ) ∩ φi (V ) = ∅ for all i 6= j.

Fact. (see [Mö])

(1) An IFS on a closed set X ⊂ Rd has a unique nonempty attractor A.

(2) If the IFS (φ1, . . . , φn) at hand is contracting ane and satises the open set condition,

then the Hausdor dimension of its attractor A is the unique solution s to the equation

n∑i=1

(ri)s = 1

where ri is the contraction factor of the map φi, for i = 1, . . . , n.

(3) For any probability vector (p1, . . . , pn) there is a unique associated self-similar measure

µ on A. If p1, . . . , pn > 0, then |µ| = A.

9.2. Ergodic CP-System generated by a single measure. The notions of mini-set,

micro-set and gallery are central in [F2]. We start with the analogous denitions in the

world of probability measures on Q. As usual, equip P (Q) with the weak-* topology induced

by C (Q).

Denition. (Mini-measures and micro-measures):

(1) Let θ ∈ P (Q). A measure ν ∈ P (Q) is called a mini-measure of θ if ν = f θ∣∣Q

(restricted to Q) for some f : Rd → Rd of the form f : x 7→ λx + u , where R 3 λ > 1

and u ∈ Rd.(2) A measure ν ∈ P (Q) is called a micro-measure of θ if there exists a sequence(νn)∞n=1

of mini-measures of ν, such that νn → ν in the weak-* topology of P (Q).

(3) A measure θ ∈ P (Q) measure is called homogeneous if any micro-measure of θ is

actually a mini-measure (that is, if the set of mini-measures of θ is weak-* closed).

(4) A measure θ ∈ P (Q) measure is called recurrent homogeneous if it is homogeneous,

and is a mini-measure of each of its own mini-measures.


Consider a general probability measure θ1 on Q. We construct a non-degenerate ergodic CP-

System that is supported on the micro-measures of θ1 (following [F2, sec. 4]) for an arbitrary

a grid-size parameter p.

Let us start with an informal outline. Recall the p-ary decomposition of Q, introduced in 5.3:

Q =⋃λ∈Λ

Qλ

where Λ = 0, . . . , p− 1d. Denote again the restriction to Qλ of the map t 7→ pt − λ by ψλ

(here, t ∈ Q and λ ∈ Λ are vectors), and dene

θλ =ψλ (θ|Qλ)θ (Qλ)

whenever this expression is dened. This is nothing new: if x ∈ Q lies in Qλ, then the mapping

T of CP-Systems gives T (θ, x) =(θλ , px− λ

). Let Ω be the smallest (weak-*) closed set in

P (Q) that contains θ1 and is invariant under all the maps θ 7→ θλ, λ ∈ Λ. Clearly, all elements

of Ω are micro-measures (of a special kind) of θ1. Since P (Q) is compact, so is Ω. Dene a

Markov operator P : C (Ω)→ C (Ω) by

(Pf) (θ) =∑λ∈Λ

θ(Qλ)f(θλ)

so that from the state θ, the Markov process will make a transition to θλ simply with probability

θ(Qλ). A distribution25 η ∈ P (Ω) is said to be stationary if it is invariant to the adjoint

operator P ∗ : P (Ω) → P (Ω), namely if P ∗η = η. Since the state-space P (Ω) is compact,

the set of stationary distributions of P ∗ is nonempty, compact, convex, and by the Krein-

Milman theorem, spanned by its extremals. This means that an extremal stationary distribution

η ∈ P (Ω) exist. In our case, if µ is the adapted measure to an extremal distribution η, then

(Φ, µ, T ) is an ergodic CP-System.

This informal outline has ignored technical issues involved. Formally, we appeal to the following

result [F2, theorem 5.1]:

Theorem. Let Q ⊂ Rd be the unit cube, and suppose that G ⊂ P (Q) is a (weak-*) closed subset,

with the property that for each θ ∈ G, all the micro-measures of θ are also in G. Then there

exists an ergodic CP-System (Φ, µ, T ), with µ adapted to a distribution ν on P (Q) supported

on G, such that for ν-almost every θ ∈ P (Q), dim |θ| = supθ′∈G dim |θ′|.

25As before, we use the term distribution for η, rather than measure, in order to avoid confusion with

elements of the state-space Ω.


Here, dim |θ| stands for the Hausdor dimension of the support of θ. This theorem allows us

to make the above arguments precise:

Corollary 9.1. Let θ1 ∈ P (Q) with dim |θ1| > 0. Then there exists a non-degenerate ergodic

CP-System (Φ, µ, T ), with µ adapted to a distribution ν on P (Q) supported on the set of all

micro-measures of θ1.

Proof. Let Q ⊂ Rd and let G ⊂ P (Q) be the smallest (weak-*) closed set in P (Q) that contains

θ1 and is invariant under the maps θ 7→ θλ for any λ ∈ Λ. Then all elements of G are micro-

measures of θ1. Invoking the theorem stated above, we obtain an ergodic CP-System (Φ, µ, T ),

with µ adapted to a distribution ν on P (Q), which is supported on G, such that for ν-almost

every θ ∈ P (Q)

(9.1) dim |θ| = supθ′∈G

dim |θ′|

> dim |θ1| > 0 .

Using the argument of 5.5, we can assume that the ergodic CP-System (Φ, µ, T ) is non-

degenerate as a CP-System on Rk for k > 1. Indeed, if the system is degenerate as a system

on the unit cube of R1, it follows that dim |θ| = 0 for ν-almost every θ, contradicting (9.1).

It is now easy to harvest results. Let θ1 ∈ P (Q) be any probability measure with dim |θ1| > 0

and contract a non-degenerate ergodic CP-System (Φ, µ, T ), supported on the micro-measures

of θ1. By Theorem 8.1, for µ-almost every (θ, x) ∈ Φ, θ has uniform continuous-scale scenery.

Let θ be such a measure. Since the system is supported on micro-measures of θ1, θ is a micro-

measure of θ1. We this nd a

Corollary. Any measure has a micro-measure with uniform continuous-scale scenery.

If, in addition, ν is homogenous, then θ1 above must be a mini-measure of θ1. Therefore we

have

Corollary. Any homogeneous measure has a mini-measure with uniform continuous-scale scenery.

Finally, if θ1 happens to be recurrent homogeneous, it is a mini-measure of each of its own mini-

measure. In particular, it is a mini-measure of its measure θ1 that has uniform continuous-scale

scenery. But the property of uniform continuous-scale scenery is clearly inherited to mini-

measures, as it is independent of scale. Since θ has uniform scaling scenery, and θ1 is a

mini-measure of θ, we conclude that θ1 has uniform scaling scenery. Concisely,


Corollary. Any recurrent homogeneous measure has uniform scaling scenery.

9.3. Self-similar measures have uniform scaling scenery. Our previous results imply

that self-similar measures have uniform continuous-scale scenery. The following result is

relatively easy and is given without proof.

Theorem 9.2. a self-similar measure on the attractor of a contracting ane IFS, satisfying

the open-set condition, is recurrent homogeneous.

Observe that by Fact 9.1, if the IFS at hand consists of two map or more, then its attractor must

have strictly positive Hausdor dimension. We therefore nally obtain the theorem stated at

the beginning of this section: a self-similar measure on the attractor of a contracting-homothety

IFS satisfying the open-set condition, has uniform continuous-scale scenery.

Theorem 9.3. Let φ1, . . . , φn : Q → Q be an iterated function system of contracting homo-

theties (Q ⊂ Rd is the unit cube and n > 2), satisfying the open set condition, with attractor

A =⋃ni=1 φi (A). Let θ be a self-similar probability measure on A corresponding to the strictly

positive probability vector (p1, . . . , pn). Then θ has uniform continuous-scale scenery.

10. Further Questions

The time constrains of this research did not allow investigation of several fascinating questions

that arise naturally in our discussion. We suggest them as possible directions for further

research.

(1) To which extent are the notions of statistical self-similarity and exact self similarity

unied under the uniform scaling scenery property?

(2) Fix a grid-size parameter p. Consider θ ∈ P (Q) and the family of all micro-measures

of θ, and the ergodic decomposition (with respect to the CP-System map T ) of the

corresponding subset of Φ. In Remark 6.12 we observed the Markovian nature of CP-

Systems implies that each ergodic component consists of micro-measures of θ and all

their corresponding points x ∈ Q. Moreover, each ergodic component is an ergodic CP-

System - and hence by Theorem 8.1, almost every measure in an ergodic component has

uniform continuous-scale scenery of the same kind. This means that we can attach to

each ergodic component just a single empirical distribution. What can we learn about


θ from this ergodic decomposition and from the empirical distribution corresponding

to each component? (Clearly, if θ has uniform scaling scenery then only one ergodic

component exists.)

(3) Let θ ∈ P (Q) be such that the map x 7→ η[θ,FR

x

]is θ-a.e. dened. It is natural to

group points x ∈ Q together, that have the same process η[θ,FR

x

]above them. This

yields a decomposition of |θ|, resembling a multi-fractal decomposition: all points x in

the same component have the same empirical distribution of the scaling scenery. What

can we learn about θ from this decomposition? When, if at all, can we disintegrate a

measure to measures with uniform scaling scenery?

(4) Theorem 8.1 tells us that we can map an ergodic CP-System (Φ, T, µ) to the corre-

sponding unique empirical distribution η. Hillel Furstenberg asked [FP] whether this

mapping is onto, that is, whether any measure-valued stationary process in continuous

time, which arises as (the) empirical distribution attached to some measure with uni-

form continuous-scale scenery, is in fact generated as the single empirical distribution

corresponding (in the manner suggested by Theorem 8.1) to some CP-System.

(5) One of the original motivations to dene continuous-scale scenery was the investigation

of the dependence of CP-System on the grid-size parameter p. It is natural to ask

whether geometric properties of generic measures in CP-Systems (such as uniform CP-

Scenery) depend on p. For example a concrete question that arises is: given a measure

θ ∈ P (Q) having uniform CP-Scenery (with respect to a certain grid-size parameter p)

does it have uniform CP-scenery with respect to another parameter q?

(6) Can the empirical distribution corresponding to a measure θ with uniform continuous-

scale scenery by used to bound or calculate the Hausdor dimension of |θ|?

Appendix: Extending a CP-System using the Rohlin Natural Extension

In this appendix, we provide a detailed proof of Theorem 7.4:

Theorem. Let (Φ, T, µ) be a non-degenerate, ergodic CP-System with grid-size parameter p.

Then there exists an ergodic extended CP-System (Φ, T , µ) with the same grid-size parameter

such that µ = πµ, where π : Φ→ Φ is the restriction map, and in fact π : (Φ, T , µ)→ (Φ, T, µ)

is a factor map.


Consider the Rohlin natural extension of the system (Φ, T, µ). The basic idea is to interpret

elements of this abstract extension as measures on the entire space Rd, and this way to obtain

the extended CP-System as a factor this natural extension.

Proof. Let (Φ, T, µ) be an ergodic CP-system with grid-size parameter p (this parameter will

be dominant but implicit in what follows). Using the general construction of the Rohlin Natu-

ral Extension, described in 2.4, we can construct an invertible measure preserving system(Φ, T , µ

), for which our system (Φ, T, µ) is a factor. By the concluding remark of 2.4,(

Φ, T , µ)is ergodic. Denote the factor map by π :

(Φ, T , µ

)→ (Φ, T, µ). This is a mor-

phism of measure-preserving systems. Note that the system(

Φ, T , µ)is given by an abstract

construction containing no explicit geometric objects, and therefore unt to serve as the de-

sired extended CP-System. We will construct the desired extended CP-System(Φ, T , µ

)with

grid-size parameter p as a factor(

Φ, T , µ), with a factor map α :

(Φ, T , µ

)→(Φ, T , µ

), such

that the diagram

(Φ, T , µ

)π

α //(Φ, T , µ

)πyysssssssssss

(Φ, T, µ)

commutes in the category of measure preserving systems.

Since π is measure-preserving, it is onto µ - mod 0. We can ignore the exceptional µ null set

and assume that π is onto Φ. Thus, it induces a bration Φ =⋃

(θ,x)∈Φ π−1(θ, x). Fix a point

(θ0, x0) ∈ Φ and choose some ϕ ∈ Φ in the ber of (θ0, x0), namely ϕ ∈ π−1 (θ0, x0). Both ϕ

and (θ0, x0) are xed in what follows. Note that the dependence of the point (θ0, x0) on ϕ is

implicit.

Let us show how the choice of ϕ ∈ π−1 (θ0, x0), the point we chose in the ber of (θ0, x0), yields

a past of the point (θ0, x0) as a sequence of elements in Φ. For n ∈ N, dene

(θ−n, x−n) = π T−n (ϕ)

(recall that T is invertible). This notation is justied by the fact that

T(θ−(n+1), x−(n+1)

)= (θ−n, x−n) .


Indeed,

T(θ−(n+1), x−(n+1)

)= T π T−(n+1) (ϕ) =

= π T T−(n+1) (ϕ) = π T−n (ϕ) = (θ−n, x−n)

where we have used the fact that, since π is a morphism of measure-preserving systems, we

have T π = π T . Again, while (θ−n, x−n) depends on ϕ, we leave ϕ implicit to simplify our

notation. This, for any choice of ϕ ∈ π−1 (θ0, x0) we obtain a two-sided sequence (θn, xn)n∈Z

extending (θn, xn)∞n=0.

In 6.1, we dened the CP-frameset FCPx =QCPx,n

n∈N around x ∈ Q by QCPx,n = Qwn(x),

where Qwn(x) is the unique cube of side-length p−n containing x in the p-ary decomposition of

Q (recall 5.3). Recall that FCPx is related to the sequence (θn, xn)∞n=0 : indeed,θCPx,n

∞n=0

(the FCPx -scenery of θ0) is just the sequence θn∞n=0. We now use the sequence with a past

(θn, xn)n∈Z , extending (θn, xn)∞n=0, to extend the frameset FCPx to a two-sided frameset.

By two-sided frameset we mean a sequence of cubes increasing to Rd and decreasing to x0.

For n ∈ Z, dene

Qn =

Qwn(x0) ≡ QCPx0,n n > 0

H⟨Qw|n|(xn) , Q

⟩(Qw|n|(xn)

)n < 0

where as usual H⟨Qw|n|(xn) , Q

⟩if the unique Rd → Rd homothety inducing a bijection of

Qw|n|(xn) onto Q.

Note that for each n ∈ Z, Qn is a cube of side-length pn containing x0. Since Qn for n > 0 is

just QCPx0,n, the sequence of cubes Qnis clearly decreasing to x0. We now claim for µ -almost

every (θ0, x0) ∈ Φ, and for any choice of ϕ ∈ π−1 (θ0, x0), the sequence Qn0n=−∞ is increasing

to the entire space, namely that⋃0n=−∞Qn = Rd.

This follows from the fact that the CP-System (Φ, T, µ) is non-degenerate. Indeed, it is easy

to verify that if Qn0n=−∞ Rd, then Qn0n=−∞ is contained in some half-space of the

form

(x1, . . . , xd)∣∣xi > 0

or

(x1, . . . , xd)∣∣xi 6 0

for some 1 6 i 6 d. By the denition

of the sequence Qn, it is clear that this can only happen if for some 0 > ` ∈ Z , x` is on

the (i , ε)-face, as dened in 5.5, of Q for ε = 0 or ε = 1. Therefore (θ`, x`) ∈ T−|`|A(|`|)i,ε

where A(|`|)i,ε is dened in (5.4). It follows that (θ0, x0) ∈ A(0)

i,ε . Since the CP-System (Φ, T, µ) is

non-degenerate, µA(0)i,ε = 0. Therefore, the event

(θ0, x0) ∈ Φ∣∣ the sequence Qn for some ϕ ∈ π−1 (θ0, x0) does not increase to Rd


is contained in theµ-null event⋃i,εA

(0)i,ε .

Let us denote the set of points ϕ ∈ Φ in the preimage π−1 (Φ), for which the sequence Qnn∈Z

does increase to the entire space Rd, by G. We have just established that µ(G)

= 1. We

proceed to dene a mapping α : G → Φ that will become the desired factor map between Φ

and Φ. Fix some ϕ ∈ G. We rst use the foregoing constructions to generate the corresponding

two-sided sequence (θn, xn)n∈Z and the two-sided frameset Qnn∈Z increasing to Rd. For

n ∈ N, set

νn =H⟨Qwn(x−n) , Q

⟩ θ−n

θ−n(Qwn(x−n)

) .

Note that νn is supported on Q−n, and that νn∣∣Q≡ θ0 (in particular, νn is normalized on

Q). In fact, νn extends νn−1 in the sense that νn∣∣Q−(n−1)

≡ νn−1. This increasing sequence

of measures converges to a limiting measure ν in the w∗-topology of M(Rd). Indeed, since⋃

n∈ZQn = Rd, for every f ∈ Cc

(Rd)the sequence

´Rd f dνn

∞n=0

is eventually constant. The

functional f 7→ limn→∞

´Rd f dνn is clearly linear and bounded, and by the Riesz representation

theorem (2.1) denes a Radon measure Rd, which we denote by θ0. It is easy to verify that

θ0

∣∣Q≡ θ0 and in fact θ0 ∈ Φ. Finally, dene α : G → Φ on our ϕby α : ϕ 7→

(θ0, x

).

We have thus dened a mapping α : Φ→ Φ on a set G of full µ-measure in Φ. Set the measure

µ on Φ by µ = α µ, which turns α :(

Φ, µ)→(Φ, µ

)into a measure-preserving map. It

follows easily from our construction that T α = α T , and therefore we have established that

α :(

Φ, T , µ)→(Φ, T , µ

)is a factor map. This also completes the construction of the desired

extended CP-System(Φ, T , µ

). As a factor of the ergodic system

(Φ, T , µ

), the extended

CP-System(Φ, T , µ

)is ergodic itself.

It remains to verify that all the maps in the above diagram are in fact morphisms of measure-

preserving systems, and that the diagram commutes. The fact that it commutes as a diagram of

functions between sets it easy: suppose that ϕ ∈ Φ. Write (θ0, x0) for π (ϕ). We have seen that

in this case α (ϕ) is a measure on Rd that extends θ0, hence π α (ϕ) = θ0 = π (ϕ). It follows

that the equality π α = π holds µ-a.e. on Φ. We can now show that π is measure-preserving:

indeed, let A ⊂ Φ be measurable. Then

µA = µ(π−1A

)= µ

(α−1

(π−1A

))= µ

(π−1A

)where we have used the facts (from left to right) that π is measure preserving, that π α = π

and that α is measure-preserving. Thus π is measure preserving. By the remark after Denition


7.3, we have π T = T π so that π too is a factor map. Therefore, the above diagram of

morphisms of measure-preserving systems commutes, as required.


List of symbols

Q = [0, 1]d the unit cube in Rd

Qx =∏di=1

[xi − 1

2 , xi + 12

]for x ∈ Rd; in particular, Q0 =

[−1

2 ,12

]dQx,t =

∏di=1

[xi − 1

2e−t , xi + 1

2e−t] in the context of continuous-scale; Qx,n =

∏di=1

[xi − 1

2p−n , xi + 1

2p−n]

in the context of p-scale.

θ probability measure on a cube Qx or (depending on context) Radon measure on Rd, normal-

ized on some cube Qx.

f θ the push forward of the measure θ by the mapping f ; f θ (A) = θ(f−1A

)|θ| the support of the measure θ

C (K) (K a compact topological space) - the space of continuous functions K → R with the

supremum norm

Cc (X) (X a locally compact topological space) the space of continuous, compactly-supported

functions X → R with the supremum norm

L1 (Ω, µ) the space of Lebesgue integrable functions on the measure space (Ω, µ)

P probability measure

P (K) The set of Borel probability measures on a compact space K with the weak-* topology

induced by C (K)

M(Rd)The set of Radon measures µ on Rd with the weak-* topology induced by Cc

(Rd)

(Denition 2.4).

All product spaces are considered with their product topology.

Λ∗ =⋃∞n=1 Λn the set of nite words over the alphabet Λ


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הפקולטה למתמטיקה ולמדעי הטבע

מכון איינשטיין למתמטיקה

:עבודת גמר לתואר מוסמך בנושא

מידות עם נוף כיול אחיד

Measures with Uniform Scaling Scenery

מתן גבישמוגש על ידי 03599852/5: מספר תלמיד

הלל פורסטנברג' העבודה הוכנה בהדרכת פרופ

16/8/2008 ח"ו באב תשס"ט

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