The Orbital Equivalence Class of Half Stationary Bernoulli Shiftsmath.huji.ac.il/~nachi/Files/Thesis_(Nachi).pdf · 2019-01-22 · stationary distribution on the non-positive coordinates

The Orbital Equivalence Class of

Half Stationary Bernoulli Shifts

Author: Nachi Avraham Re'em

Supervisor: Prof. Zemer Koslo

January 2018

A thesis submitted for M.Sc in mathematics

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

1

Introduction

The non-singular ergodic theory is usually engaged with a transformations Tof a sigma-nite standard Borel space (X,B, µ). The transformation is notnecessarily measure-preserving but yet is non-singular; that is, the inverse imageof every null measure set is a null measure set.Beyond the natural questions such as ergodicity and conservativity of T , we areinterested in the following question: Is there a sigma-nite Borel measure whichis both equivalent to µ and invariant to T? We call such measure a.c.i.m. (abso-lutely continuous invariant measure). The interest of such measure arises fromorbital-equivalence theory as we will explain later. We say that a transformationis of type II if there exists a.c.i.m. or of type III otherwise. Halmos askedwhether there exists type III transformations at all and it has been answeredpositively by Ornstein [Orn].1 An example of a Bernoulli shift of type III hasbeen given later by Hamachi [Ham].One of the very fundamental properties of this classication is that it is an invari-ant of orbital-equivalence. We will introduce the notion of orbital-equivalenceand the celebrated theorem by Dye asserts that all ergodic transformation oftype II where the corresponding a.c.i.m. is nite are orbit-equivalent. Thisorbit-equivalence class is called type II1. Krieger introduced the ratio setfor a non-singular transformation and showed that it is an invariant of orbit-equivalence. The ratio set decomposes the ergodic non-singular type III trans-formations further into subclasses IIIλ, 0 ≤ λ ≤ 1, and Krieger showed that foreach 0 < λ ≤ 1, the transformations which are type IIIλ are all orbit-equivalent.He further classify the type III0 transformations but this will not be consideredhere. For further reading see [HOO] and [KW]. We will study and use thefollowing fact for this classication: Type III1 transformation are exactly theergodic non-singular transformation with ergodic Maharam extension.The question what are the possible types IIIλ for a Bernoulli shift was raisedby Krieger and Weiss and answered partially by Koslo [Kos]: He observed

that the Bernoulli shift over 0, 1Z with respect to a uniform half-stationarymeasure, that is measure which is uniform in every non-positive coordinates, isthe natural extension of the one-sided shift over 0, 1N with the same measureon the positive coordinates. Using this, he showed that a conservative half-stationary Bernoulli shift of type III is of type III1 and in any other case it is oftype II1. Danilenko and Lemanczyk [DL] extended this result for 0, 1Z to any

1We should mention here that it turned out that the positive answer to this question hasbeen given earlier by von Neumann in 1940. In the late '30s von Neuman and Murray publisheda series of papers establishing the subject which is now called von Neumann algebras. Somevon Neumann algebras are called factors and classied by type I, II and III. It was not clearwhether there exists type III factor and a paper of von Neumann was dedicated to constructan example of such factor [vonN]. It turned out that every commutative factor arises froma non-singular transformation. Moreover, two factors are of the same type if and only if thecorresponding non-singular transformations are orbit-equivalent. We refer to the survey [DS,11.8]. We will also mention that the example of type III given by von Neuman, arises fromthe 2-adic odometer with probabilities (p, 1− p) where p 6= 1/2.

2

stationary distribution on the non-positive coordinates. In this work we provethe more general assertion that a conservative shift over any nite state spaceand every half-stationary measure is either of type II1 or of type III1.We think that our proof is exible enough for more complicated models asinhomogeneous Markov subshifts, which arises from the study of a natural classof hyperbolic toral automorphisms.

3

CONTENTS CONTENTS

Contents

Contents 4

1 Preliminaries 5

I Basics of Non-Singular Ergodic Theory 61.1 Conservative Transformations . . . . . . . . . . . . . . . . . . . . 61.2 Ergodic Transformations and the K-Property . . . . . . . . . . . 10

2 Skew-Products and Maharam Extension 13

3 Orbital Equivalence and the Ratio Set 16

3.1 The Associated Flow . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 The Orbital Equivalence Relation 28

5 Kakutani's Criterion 33

6 Future Research: Inhomogeneous Markov Subshift 36

II Half-Stationary Bernoulli Shift is of Type II1 or III1 40

References 49

A Appendix: Paley-Zygmund Inequality 51

4

1 PRELIMINARIES

1 Preliminaries

Let S be a nite state space and let X = SZ. Endow X with the Borel sigma-algebra B generated by the cylinder sets, namely sets of the form

[~a]N+MN = x ∈ X | xN = aN , . . . , xN+M = aN+M

for some integers N,M and ~a ∈ SM+1. The collection of nite unions of cylindersets, the sets supported on nitely many coordinates, forms an algebra of sets.This collection is also a basis for the metric topology on X induced by

n (x, y) := 2− infn∈N|xn 6=yn

We can think of the sigma-algebra as generated by the functions X → R thatdepend on nitely many coordinates. The space (X,B) is a standard Borel spacewhere the family of the cylinders is a basis of clopen sets for the topology.Let T : X → X be the shift map dened by (Tx)n = xn+1 for every n ∈ Z.This is a Borel automorphism of X onto itself. We will consider Borel productmeasures on X, measures of the form µ =

∏n∈Z µn, where µn is a distribution

on S such that µ(

[a]N+MN

)=∏N+Mn=N µn (an) for every cylinder. This denes a

measure uniquely by Kolmogorov's extension theorem. The system (X,B, µ, T )is called Bernoulli shift or Bernoulli scheme.A product measure µ =

∏n∈Z µn is called half-stationary if for some dis-

tribution ~p over S we have µn = ~p not depending on n for every n ≤ 0. Incase of a half-stationary product measure, T has the one-sided Bernoulli shift,that is the shift σ of the space X+ = SN with the usual Borel sigma-algebraF and the product measure µ+ =

∏n∈N µn, as a markovian factor; that is,

a factor that carries Radon-Nikodym derivative to Radon-Nikodym derivative.Moreover, T is a natural extension of σ and it is unique if we add the propertythat the Radon-Nikodym derivative is measurable with respect to the positivecoordinates. Notice that by Kolmogorov 0 − 1 law, since the coordinates areindependent with respect to product measures, σ is exact, that is

⋂n∈N σ

−nFis trivial mod µ+. The fact that also the Radon-Nikodym derivative of theshift T is measurable with respect to the positive coordinates, means that T hasK-property as we will dene later.The main object in this work is the dynamics of the shift T : X → X over thestandard Borel space (X,B, µ), where µ is a half-stationary product measure.

Remark. In general, we write A = B if µ (A4B) = 0 for every Borel sets A,B.

5

Part I

Basics of Non-Singular Ergodic

Theory

Some of the following discussion can be found in Aaronson's textbook of non-singular ergodic theory [Aar, Chapter 1.1].

Let (X,B, µ) be a standard Borel space where µ is sigma-nite Borel measure.From now on we will discuss only sigma-nite measures. We say that a measureν on B is equivalent to µ, if each of them is absolutely continuous with respectto the other; that is, ν (A) = 0 ⇐⇒ µ (A) = 0 for every A ∈ B. In this case wewrite µ ∼ ν. A measurable transformation T : X → X is called non-singularif µ ∼ µ T .For an invertible non-singular transformation, there exists the Radon-Nikodymderivative

T ′ :=dµ Tdµ

∈ L1 (X,B, µ)

which is positive a.e.. The non-singularity of T implies the non-singularity ofTN for every N ∈ Z, so we denote

(TN)′

:=dµ TN

dµ∈ L1 (X,B, µ)

The basic properties of Radon-Nikodym derivative yields the relation,(TN+M

)′=(TN)′ · (TM)′ TN

for every N,M ∈ Z.

1.1 Conservative Transformations

Let T be a probability measure preserving transformation. Poincare's recurrencetheorem asserts that

∑∞n=1 1A Tn = ∞ a.e. on A, for every positive measure

set A. This property is what we will call in a wider context as Halmos' conditionfor conservativity. We start by denition of conservative part and dissipativepart:

Denition. A measurable set W ∈ B is called wandering if the elements ofthe collection

T−1W,T−2W,T−3W, . . .

are pairwise disjoint. Let D (T ) ⊂ X be the union of all the wandering sets inX. Then D (T ) is called the dissipative part of X with respect to T . Notethat the collection of all the wandering sets is hereditary (that is, closed undertaking subsets). Thus by the exhaustion lemma (see [Aar, Lemma 1.0.7]) its

6


union set D (T ) is a countable union of wandering sets and in particular D (T )is measurable.Let C (T ) := X\D (T ). Then C (T ) is called the conservative part of X. Onecan see that C (T ) and D (T ) are T -invariant sets. We say that T is conserva-tive if X = C (T ) and that T is dissipative if X = D (T ).

Remark. We note that for an invertible and non-singular transformation T , forevery positive measure set W we have by non-singularity that the elements ofT−nWn∈N are pairwise disjoint if and only if the elements of TnWn∈N arepairwise disjoint. Thus, the collection of T -wandering sets is identical to thecollection of T−1-wandering sets, so D (T ) = D

(T−1

)and C (T ) = C

(T−1

).

We mention now Halmos' condition which one can use to show that C(TN)

=C (T ) for every N ∈ N. In general we get that for an invertible and non-singulartransformation T , C

(TN)

= C (T ) for every N ∈ Z.

Theorem (Halmos' condition [Aar, Theorem 1.1.1]). Let T be a non-singulartransformation of a standard Borel space (X,B, µ). Then A with µ (A) > 0 iscontained in C (T ) if and only if

∞∑n=1

1A Tn =∞ a.e. on A

Theorem (Maharam's condition). Let T be a non-singular transformation of astandard Borel space (X,B, µ). Then T is conservative if and only if for everyset A with µ (A) > 0,

T−1A ⊂ A =⇒ T−1A = A

This property is called incompressibility.

Proof. Notice that if T−1A ⊂ A then W := A\T−1A is a wandering set, henceif T is conservative and T−1A ⊂ A then T−1A = A. For the other direction,if there exists a wandering set W with µ (W ) > 0, then A :=

⋃n≥0 T

−nWcontradicts the incompressibility.

Remark. Notice that by T -invariance of C (T ), we can reduce the discussionto the conservative part

(C (T ) ,B ∩ C (T ) , µ |C(T )

)and the above assertions

characterize the conservative part C (T ) inside X.

Theorem (Hopf's Decomposition). Let T be a non-singular transformation ofa standard Borel space (X,B, µ). Then:

1. In the probability case:

C (T ) =

∞∑n=1

(Tn)′

=∞

2. In the measure preserving case:

C (T ) =

∞∑n=1

f Tn =∞

7


For every f ∈ L1 (µ) with f > 0. If instead of f > 0 we have only f ≥ 0,then

C (T ) ⊃

∞∑n=1

f Tn =∞

Proof.

1. We show equivalently that

D (T ) =

∞∑n=1

(Tn)′<∞

Denote ∞∑

n=1

(Tn)′<∞

=

∞⋃N=1

AN , AN :=

∞∑n=1

(Tn)′< N

For every N ∈ N, by bounded convergence we haveAN

(∑∞n=1 (Tn)

′(x))dµ (x) =

∑∞n=1

AN

(Tn)′(x) dµ (x)

=∑∞n=1µ (TnAN ) =

(∑∞n=11TnAN (x)) dµ (x)

=

(∑∞n=11AN

(T−nx

))dµ (x)

The left side is bounded by N · µ (AN ) hence nite, so it follows that

∞∑n=1

1AN T−n <∞ a.e. on X

and we see by Halmos' condition that µ(AN ∩ C

(T−1

))= µ (AN ∩ C (T )) =

0. But N is arbitrary so we see that

D (T ) ⊃

∞∑n=1

(Tn)′<∞

On the other hand, using Halmos' condition for every wandering set W ,

∞∑n=1

1W T−n <∞

so by the same calculation as in the rst part of the proof we see that

W

( ∞∑n=1

(Tn)′(x)

)dµ (x) =

( ∞∑n=1

1W(T−nx

))dµ (x) <∞

Hence∑∞n=1 (Tn)

′(x) <∞ a.e. on W and we get

D (T ) ⊂

∞∑n=1

(Tn)′<∞

8


2. Let f ∈ L1 (µ) with f ≥ 0. Let W be a wandering set, that is∑∞n=1 1W

Tn ≤ 1. We see that for every N ≥ 1,

W

(∑Nn=1f T

n)dµ =

∑Nn=1

Wf Tndµ =

∑Nn=1

Wf TN−ndµ

=∑Nn=1

1W Tn · f TNdµ

=

(∑Nn=11W Tn

)· f TNdµ

≤f TNdµ =

fdµ ≤ ‖f‖1 <∞

hence we get∞∑n=1

f Tn <∞ a.e. on W

so we conclude

C (T ) ⊃

∞∑n=1

f Tn =∞

Assume now that f > 0 strictly. Let A ⊂ C (T ) with µ (A) > 0. Denote

A =

∞⋃N=1

AN , AN := A ∩ f ≥ 1/N

For every N , AN ⊂ A ⊂ C (T ), so by Halmos' condition we see that

∞∑n=1

f Tn ≥ 1

N

∞∑n=1

1AN Tn =∞ a.e. on AN

But N is arbitrary so

∞∑n=1

f Tn =∞ a.e. on A

and we conclude

C (T ) ⊂

∞∑n=1

f Tn =∞

Corollary (Maharam's Recurrence Theorem). Let T be a measure preservingtransformation of a standard Borel space (X,B, µ). If there exists some A ∈ Bwith µ (A) <∞ such that

X =

∞⋃n=0

T−nA

Then T is conservative.

9

1.2 Ergodic Transformations and the K-Property

Proof. By the assumption we see that for every N ≥ 1,

X = T−NX =

∞⋃n=N

T−nA

Hence∞∑N=1

1A TN =∞ a.e. on X

The theorem now follows from part 2 of Hopf's decomposition for f = 1A.

We introduce now some kind of generalization of Hopf's theorem. In order todene it we dene rst the dual operator. For more see [Aar, Chapter 1.3].

Denition. Let T be a non-singular transformation of a standard Borel space

(X,B, µ). We dene the dual operator T : L1 (µ)→ L1 (µ) by T f :=dνfT−1

dµ ,where dνf = fdµ.It is called dual since it is the dual of the Koopman operator g 7→ g T forg ∈ L∞ (µ), in the sense that

T f · gdµ =

f · g Tdµ for every f ∈ L1 (µ)

and g ∈ L∞ (µ).

Note that if T is invertible then T f = T ′ · f T−1.

Theorem (Hopf's Decomposition general version). Let T be a non-singulartransformation of a standard Borel space (X,B, µ). Then

C (T ) =

∞∑n=1

Tnf =∞

for every f ∈ L1 (µ) with f > 0.


Denition. We say that a non-singular transformation T of a sigma-nite space(X,B, µ) is ergodic if whenever T−1A = A, either µ (A) = 0 or µ (X\A) = 0.The collection I :=

A ∈ B | T−1A = A

is a sub-sigma-algebra of B which is

trivial mod µ exactly when T is ergodic. A standard argument shows that er-godicity is equivalent to the fact that every T -invariant function of L∞ (X,B, µ)is a.e. constant.

Proposition. Let T be a non-singular and conservative transformation of astandard Borel space (X,B, µ). Then T is ergodic if and only if

∑∞n=1 fTn =∞

a.e. on X for every f ∈ L1 (X,B, µ) with f ≥ 0, f 6= 0.

Proof. It follows directly by the fact that a set of the form ∑∞n=1 1A Tn =∞

is T -invariant and contains A (by Halmos' condition for conservativity). Thegeneralization for every f is a standard argument.

10


We mention here shortly the pointwise ergodic theorem for non-singular trans-formations. For a measure-preserving transformation over sigma-nite spacesthere is the ratio ergodic theorem due to Hopf. We introduce here the general-ization for non-singular transformations due to Hurewicz. A detailed discussioncan be found in [Aar, Chapter 2.2].

Recall that the dual operator T : L1 (X,B, µ) → L1 (X,B, µ) is dened by

T f :=dνfT−1

dµ , where dνf = fdµ. For the dual operator we haveT f · gdµ =

f · g Tdµ for every f ∈ L1 (µ) and g ∈ L∞ (X,B, µ). Also if T is invertible

then T f =(T−1

)′ · f T−1.

Theorem (The Hurewicz's Ergodic Theorem). Let T be a non-singular andconservative transformation of a standard Borel space (X,B, µ). Then for everyf, p ∈ L1 (X,B, µ) with p > 0, the following limit exists a.e. on X,

h [f, p] := limN→∞

f + T f + · · ·+ TNf

p+ T p+ · · ·+ TNp

Remark. In general it is shown that this limit is h [f, p] = Eνp

[fp | I

]where

dνp = pdµ. Hence h [f, p] is T -invariant function with the propertyA

h [f, p] ·pdµ =

Afdµ for every A ∈ I. In particular, if T is ergodic then h [f, p] is

constant and,

h [f, p] =

fdµpdµ

a.e. on X

Denition. A non-singular transformation (Y,F , ν, S) is called an extensionof some non-singular transformation (X,B, µ, T ), and the latter is called a fac-tor of the former, if there exists a map π : Y → X for which:

1. π−1B ⊂ F

2. π∗ν ∼ µ (Note that this is non equality but only equivalence.)

3. π S = T π

In this sense, two non-singular transformation are isomorphic, if the map π isinvertible and those three conditions hold for π−1 as well.

Denition. Suppose that (X,B, µ, T ) is a non-singular transformation. We saythat an extension (Y,F , ν, S) of this is a natural extension, if S is invertibleand for the factor map π : Y → X we have,

∞∨n=1

Snπ−1B = F

where∨

denotes the generated sigma-algebra.

Remark. It is well-known that the natural extension of measure-preservingtransformation exists and is unique up to isomorphism. In the non-singularcase we need further to consider what is called markovian extension to getuniqueness. See [ST, Theorem 4.2].

11


Denition. [ST, Denition 4.5 and Remark 4.6.b] Let T be an invertible non-singular transformation of a standard Borel space (X,B, µ). We say that Tsatises the K-property (K for Kolmogorov), if there exists a sub-sigma-algebra A ⊂ B such that the following hold:

1. T−1A ⊂ A

2.⋂n∈Z T

nA is trivial mod µ.

3.∨∞n=0 T

nA = B

4. T ′ = dµTdµ is A-measurable.

Notice that conditions 1-3 are the usual denition of K-automorphism in thecontext of measure preserving transformations, while condition 4 comes to en-sure that the natural extension is unique up to isomorphism of non-singulartransformations.

Denition. A non-singular transformation T of a standard Borel space (X,B, µ)is called exact, if

⋂∞n=1 T

−nB is trivial mod µ.

Remark. If T is an invertible non-singular transformation of a standard Borelspace (X,B, µ) with K-property with respect to A, then it is the natural exten-sion of the exact transformation T of (X,A, µ |A).

Example (Kolmogorov's 0−1 law). Let S be a nite set and consider the one-sided shift σ over the Borel space

(SN,B, µ

), where µ is some product measure

with no zero probabilities. Then by Kolmogorov's 0 − 1 law σ is exact henceergodic with respect to µ.The half stationary Bernoulli shift has K-property with respect to the sub-sigma-algebra of the positive coordinates. Conditions 1 and 3 are trivial. Condition 2is Kolmogorov 0−1 law. Condition 4 follows as one can verify on the monotoneclass of the cylinders that the Radon-Nikodym derivative is

dµ Tdµ

(x) =

∞∏n=1

µn−1 (xn)

µn (xn)

Proposition. A non-singular and invertible transformation T of a standardBorel space (X,B, µ) with K-property with respect to A, is either dissipative orconservative. Moreover, if T is conservative then it is also ergodic.

Remark. Here we include a proof for the ergodicity in the probability measurecase. For the proof of the general case, we refer to [ST, Lemma 4.3] where ithas been established that if

∨∞n=1 T

nA = B and f/fT is A-measurable for fnon-negative, then f is also A-measurable.

Proof. The dichotomy is a direct result of the Hopf's decomposition: Identify

D (T ) =∑∞

n=1 Tnf <∞

for some f ∈ L1 (A) with f > 0. Since T ′ and f

are A-measurable so is T f =(T−1

)′ · f T−1 and so is D (T ). Since D (T ) is

12

2 SKEW-PRODUCTS AND MAHARAM EXTENSION

T -invariant set it is measurable with respect to⋂∞n=1 T

−nA, hence it is eithernull or full measure.As for the ergodicity we consider the probability measure case. Recall that byHurewicz's ergodic theorem, conservativity implies that for every f ∈ L1 (X,B, µ),∑N−1

n=0 Tnf∑N−1

n=0 Tn1

L1

−−−−→N→∞

E [f | I]

where I is the sigma-algebra of T -invariant Borel sets. Fix a monotonic sequence

fkL1

−−−−→k→∞

f where fk is∨∞n=−k T

nA-measurable. For every k ∈ N, h [fk, 1] is⋂∞n=1 T

−nA-measurable hence by exactness of A we have that h [fk, 1] =fkdµ

is constant. Using the monotonicity of conditional expectation we see that

E [f | I] = limk→∞

E [fk | I] = limk→∞

fkdµ =

fdµ

is constant. It follows that E [f | I] =fdµ is constant for every f ∈ L1 (µ)

hence T is ergodic.

2 Skew-Products and Maharam Extension

Let T be a non-singular transformation of a standard Borel space (X,B, µ). Afunction ϕ : N×X → R or ϕ : Z×X → R is called a R-valued cocycle, if fora.e. x ∈ X and every N,M ,

ϕN+M (x) = ϕN (x) + ϕM(TNx

)Given any function ϕ : X → R we can dene a cocycle by

ϕN := ϕ+ ϕ T + · · ·+ ϕ TN−1

For a cocycle ϕ, we dene the skew-product extension of (X,B, T, µ) by

Tϕ : (X × R,BX ⊗ BR, ν)

where ν = µ⊗m for some m measure on R and,

Tϕ (x, t) := (Tx, t− ϕ (x))

Denition. [Mah, 3.1] The Maharam extension T of an invertible non-singular transformation T is the skew-product by the Radon-Nikodym cocycleand the measure ν = µ⊗ exp (s) dx. That is, T is the transformation

T (x, t) := (Tx, t− log T ′ (x))

Claim. The Maharam extension is measure-preserving.

13


Proof. Consider the sub-basis of sets of the form E × (−∞, a]. We see that

T−1 (E × (−∞, a]) =

(x, t) ∈ X × R | x ∈ T−1E, t < a+ log T ′ (x)

so by Fubini theorem we get,

ν(T−1 (E × (−∞, a])

)=

T−1E

( a+log T ′(x)

−∞ exp (s) ds)dµ (x)

=T−1E

T ′ (x) dµ (x) · exp (a)

= µ (E) · exp (a) = ν (E × (−∞, a])

It follows that there exists a p-system Π ⊂ BX ⊗BR that generates BX ⊗BR forwhich ν |Π= νT−1 |Π. From Dynkin's p-l theorem ν = νT−1 on BX⊗BR.

Theorem (Maharam [Mah, Theorem 5.1]). T is conservative if and only if Tis conservative.

In order to prove this we need the following general facts concerning conservativetransformation. Both can be found in [Mah, Lemma 5.2 and 5.3].

Proposition (Halmos-Ornstein). Let T be an invertible, non-singular and con-servative transformation. Then

µ T ≤ µ =⇒ µ T = µ

Proof. If µ T 6= µ then there exists measurable set E with µ (TE) < µ (E).Dene for every x ∈ E,

nE (x) := min n ∈ N | Tnx ∈ E

this is a measurable mapping since n−1E (k) = T−kE\

⋃k−1i=0 T

−iE. Also byconservativity nE < ∞ a.e. in E. Write E as disjoint union E =

⊎∞k=1Ek,

where Ek := E ∩ nE = k and let

F :=

∞⋃k=1

k−1⋃i=0

T iEk

We see that

TF =

∞⋃k=1

k⋃i=1

T iEk =

∞⋃k=1

k−1⋃i=1

T iEk ∪∞⋃k=1

T kEk ⊂ F ∪ E = F

and that

F\TF = E\∞⋃k=1

T kEk =

∞⊎k=1

(Ek\T kEk

)Since µ (TE) < µ (E) there exists k0 with µ

(T k0Ek0

)< µ (Ek0) , and in partic-

ular µ(Ek0\T k0Ek0

)> 0. It follows that

µ (F\TF ) =

∞∑k=1

µ(Ek\T kEk

)> 0

14


that is TF 6= F , which means that T is not incompressible hence not conserva-tive, a contradiction.

Proposition (Dual incomplressibility). Let T be an invertible, non-singular andconservative transformation. Then for every non-negative measurable functionf : X → R,

T−1f ≤ f =⇒ T−1f = f

where T g =(T−1

)′ · g T−1 is the dual operator.

Proof. Assume by contradiction that there exists some non-negative measurablefunction f : X → R with (1) f T · T ′ ≤ f and (2) µ (P ) > 0 where P :=T ′ · f T < f.Step 1 is dedicated to nd a measurable set Y ⊂ X with µ (Y ) > 0 such thatP ⊂ Y and f |Y is positive and nite. This will allow us in step 2 to denethe system (Y,B ∩ Y, µfY , T |Y ) where µfY = f |Y dµ, for which we will useHalmos-Ornstein theorem.

1. Let

O := f = 0 , O :=

∞⋃n=0

T−nO

By assumption (1) we see that TO ⊂ O so by incompressibility TO = O.Also TO ⊂ O so again by incompressibility TO = O. It follows thatO = O and we conclude that P ∩O = P ∩O = ∅.Let

A := 0 < f <∞ ∩(X\O

)Since TP ⊂ A and µ (TP ) > 0 by non-singularity, also µ (A) > 0. Byassumption (1) we have TA ⊂ A so by incompressibility TA = A.

Let

Y :=

∞⋂n=1

TnA

We have that Y ⊂ TA = A. Also A\Y = TA\Y = ∅ so A = Y and inparticular µ (Y ) > 0. Also since T is invertible and TP ⊂ A = TA = TY ,we see that P ⊂ Y .

2. Now consider the restrictions f |Y and T |Y . Clearly assumption (1) stillholds, and by P ⊂ Y we see that (2) holds as well. Also by the denitionof A and the fact that A = Y we have that 0 < f |Y<∞ a.e. on Y .

Consider the measure µf |Y := f |Y dµ. Then µf |Y ∼ µ |Y and by assump-tion (1) we see that µf |Y T ≤ µf |Y . Thus by Halmos-Ornstein theorem

for(Y,B ∩ Y, µf |Y , T |Y

)we get that µf |Y T |Y = µY .

Now we can get a contradiction: Fix some 0 < s < t such that P ′ = P ∩s < f |Y< t is of positive measure. Since P ′ ⊂ P we see by the denition ofP that µf |Y (TP ′) < µf |Y (P ′), a contradiction.

15

3 ORBITAL EQUIVALENCE AND THE RATIO SET

Proof. (Maharam's theorem). If T is not conservative then there exists a wan-dering set W ∈ B with µ (W ) > 0. It follows that W × R ⊂ X is a wanderingset for T , so X is not conservative. Assume now that T is conservative. DeneF : X → R by

F (x) := supn≥0

(Tn)′(x)

By (Tn)′

= T ′ ·(Tn−1

)′ T we see that(T−1

)′ · F = supn≥0

(Tn−1

)′ T = supn≥−1

(Tn)′ T ≥ sup

n≥0(Tn)

′ T = F T

so by dual incompressibility (T−1

)′ · F = F T

Let a ∈ R. We see that ν (X × (−∞, a]) = exp (a) <∞. Dene

Ea :=

∞⋃n=0

T−n (X × (−∞, a]) = (x, t) ∈ X × R | t < a+ logF (x)

and we see that

T−1Ea = (x, t) ∈ X × R | (Tx, t− log T ′ (x)) ∈ Ea= (x, t) ∈ X × R | t < a+ logF (Tx) + log T ′ (x)= (x, t) ∈ X × R | t < a+ logF (x) = Ea

It follows that Ea =⋃∞n=0 T

−nEa and µ (Ea) < ∞, hence by Maharam's re-

currence theorem T |Ea is conservative, that is Ea ⊂ C(T). Since a ∈ R is

arbitrary we see that T is conservative.

3 Orbital Equivalence and the Ratio Set

The basics of the orbital-equivalence and the ratio set are from [Sch].Let (X,B, µ) be a standard Borel space where µ is sigma-nite Borel measure.The family of all invertible non-singular transformations of (X,B, µ) to itselfforms a group with respect to composition. This group is called the automor-phisms group of (X,B, µ) and is denoted by Aut (X,B, µ).Let G be a countable group. An action TG : Gy (X,B, µ) is a homomorphismfrom G to the automorphism group Aut (X,B, µ). We denote such action byg 7→ Tg. We also denote TGx := Tgx | g ∈ G for x ∈ X and TGA :=

⋃g∈G TgA

where TgA := Tgx | x ∈ A for a Borel set A ∈ B.For an action TG : Gy (X,B, µ) we associate the full group,

[TG] := V ∈ Aut (X,B, µ) | ∀x ∈ X ∃g ∈ G, V x = Tgx

16


We say that two actions T(i)Gi

: Gi y (Xi,Bi, µi), i = 1, 2, are orbit-equivalentif there exists a Borel isomorphism of non-singular transformations φ : (X1,B1, µ1)→(X2,B2, µ2) such that

φ(T

(1)G1x)

= T(2)G2φ (x)

as sets, for a.e. x ∈ X1.A celebrated theorem by Dye, for action of G = Z, and a later theorem byOrnstein andWeiss for any action of amenable groupG, asserts that there is onlyone orbital-equivalence class of non-atomic measure-preserving ergodic G-actionon a probability space. In particular every such action is orbital-equivalent to aZ-action generated by some ergodic measure-preserving transformation T . Thisis what we will call later type II1. For more details see [KM].While the probability-preserving ergodic case contains only one type as Dye'stheorem shows, Krieger further showed that the non-singular case is muchmore rich: He showed that type III transformations contain a unique orbital-isomorphism class for every 0 < λ ≤ 1, each is called type IIIλ, and this classi-cation is determined by the ratio set which we will dene soon. This notion ofratio set is useful for us and will be discussed in this chapter.In the following we restrict ourselves to discuss G = Z-actions. For this case,with a single transformation T generating a Z-action, we write the full group as

[T ] =V ∈ Aut (X,B, µ) | ∃φ : X → Z such that V x = Tφ(x)

Denition. Let T be a non-singular transformation of a standard Borel space(X,B, µ). We say that r ∈ R is an essential value for a T -cocylce ϕ, if forevery E ∈ B with µ (E) > 0, for every ε > 0 there exists n ∈ Z such that

µ(E ∩ T−nE ∩ |ϕn − r| < ε

)> 0

In other words, it means that for every E ∈ B with µ (E) > 0, for every ε > 0there exists F ⊂ E with µ (F ) > 0 and T−nF ⊂ E for some n, such that|ϕn |B −r| < ε.The set of all essential values of ϕ is called the ratio set and is denoted bye (T, ϕ).

Claim. The ratio set e (T, ϕ), if not empty, is a closed subgroup of R.

Proof. It is clear that e (T, ϕ) is a closed set. To show that e (T, ϕ) is a subgroup,let r, s ∈ e (T, ϕ) and we show that r − s ∈ e (T, ϕ).Let E ∈ B with µ (E) > 0 and ε > 0. From s ∈ e (T, ϕ) there exists n such that

µ(E ∩ T−nE ∩ |ϕn − s| < ε/2

)> 0

Denote this positive measure set by F . From r ∈ e (T, ϕ) there exists m suchthat

µ(F ∩ T−mF ∩ |ϕm − r| < ε/2

)> 0

17


By the denition of F we see that

µ(F ∩ T−mF ∩ |ϕn − s| < ε/2 ∩ |ϕm − r| < ε/2

)> 0

Note that ϕm−n Tn = ϕm − ϕn and F ⊂ T−nE, so

0 < µ(F ∩ T−mF ∩ |ϕm−n Tn − (r − s)| < ε

)≤ µ

(T−nE ∩ T−mE ∩ |ϕm−n Tn − (r − s)| < ε

)By non-singularity of Tn we conclude that

µ(E ∩ Tn−mE ∩ |ϕm−n − (r − s)| < ε

)> 0

and the proof is complete.

Corollary. The ratio set e (T, ϕ), if not empty, is either discrete lattice of theform |log λ|Z where 0 < λ ≤ 1, or the whole R.This follows from a general fact: Every additive subgroup of R is either discretelattice or dense in R. The former case occurs when the subgroup contains apositive element closest to zero. As for the other case, we see that the subgroupcontains a Cauchy sequence (near the inmum over the positive elements), thusit contains elements arbitrarily close to zero. The density follows now directly.

Claim (Choksi-Hawkins-Prasad [CHP]). . If r ∈ e (T, ϕ), then for every E ∈ Bwith µ (E) > 0 and ε > 0,

E =⋃n∈Z

E ∩ T−nE ∩ |ϕn − r| < ε

Proof. There exists n1 ∈ Z such that µ (E ∩ T−n1E ∩ |ϕn1− r| < ε) > 0.

Denote this positive measure set by F1. Letting F1 be maximal we see that F1∩Tn1F1 = ∅, F1 ∪ Tn1F1 ⊂ E and |ϕn1 |F1 −r| < ε. Put E2 := E\ (F1 ∪ Tn1F1)and by the same reasoning there exists n2 ∈ Z and F2 ⊂ E2 with µ (F2) > 0such that F2 ∩ Tn2F2 = ∅, F2 ∪ Tn2F2 ⊂ E2 and |ϕn2

|F2−r| < ε. In this

way we get a sequence of positive measure sets F1, F2, . . . and a sequence ofintegers n1, n2, . . . , both are possibly nite, such that Fk ∪ TnkFk ⊂ E and theintersections Fk ∩ Fl, Fk ∩ TnlFl, TnkFk ∩ TnlFl are all empty for every k 6= l.By the property of r ∈ e (T, ϕ) we can exhaust E such that the set E\

⋃∞k=1 (Fk ∪ TnkFk)

does not contain any positive measure set.

Corollary. Dene ϕV := ϕφ for V = Tφ ∈ [T ]. Note that ϕV (x) = ϕ (φ (x) , x)depends on x in both coordinates. We see that the cocycle property impliesϕV V ′ = φV V ′+ϕV ′ for every V, V

′ ∈ [T ]. The proof of the above claim showsthat if r ∈ e (T, ϕ), then for every E ∈ B with µ (E) > 0 and ε > 0 we can deneV = Tφ ∈ [T ] by φ (x) = nk for x ∈ Fk ∪ TnkFk and φ (x) = 0 otherwise. Forthis V we have V E = E and |ϕV − r| < ε a.e. on E. That is,

E = E ∩ V −1E ∩ |ϕV − r| < ε

18


It is not always easy to determine whether a number is an essential value, espe-cially because it has to be veried for every Borel set. The following conditionallows us to consider only a dense family of sets:

Lemma (Choksi-Hawkins-Prasad [CHP, Lemmas 2.1-2.2]). Let T be a non-singular transformation of a standard Borel space (X,B, µ). Suppose C ⊂ B is aµ-dense family of sets, with respect to the metric η (E,F ) = µ (E4F ). Let r ∈R. Then the following condition is necessary and sucient to r ∈ e (T, log T ′):

For every ε > 0 there exists δ > 0 such that for every C ∈ C withµ (C) > 0,

µ

(⋃n∈Z

(C ∩ T−nC ∩

∣∣log (Tn)′ − r

∣∣ < ε))

≥ δµ (C)

Proof. On one hand, if r ∈ e (T, log T ′) then the condition holds for every δ ≤ 1by the claim above. Assume that the condition holds for some 0 < δ < 1. LetE ∈ B with µ (E) > 0 and ε > 0. Fix some C ∈ C such that µ (E4C) < 1

2δµ (E)and by the assumption we have

µ

(⋃n∈Z

(C ∩ T−nC ∩

∣∣log (Tn)′ − r

∣∣ < ε))

≥ δµ (C)

By the corollary above there exists V = Tφ ∈ [T ] corresponding to C such that

µ(C ∩ V −1C ∩ |log V ′ − r| < ε

)≥ δµ (C) ≥ δµ (C ∩ E)

≥ δ(

1− 1

2δ

)µ (E)

DenoteD = C∩V −1C∩|log V ′ − r| < ε and we show that µ(D ∩ E ∩ V −1E

)>

0, which means that µ(E ∩ V −1E ∩ |log V ′ − r| < ε

)> 0 so r ∈ e (T, log T ′).

Denote F = D ∩ E. By the non-singularity of V it is enough to show thatµ (V F ∩ E) > 0. By the assumption log V ′ > r − ε on D, we see that

µ (V F ) =

F

V ′dµ ≥ exp (r − ε)µ (F )

≥ exp (r − ε)µ (C ∩ E) ≥ exp (r − ε)(

1− 1

2δ

)µ (E)

Note also that V F ⊂ C so µ (V F ∩ E) > 0, as required.

Proposition. If µ ∼ ν then

e

(T, log

dµ Tdµ

)= e

(T, log

dν Tdν

)

19


Proof. We show that ⊂ and the other direction is symmetric. Denote f :=log dν

dµ . Notice that for every n we have by the basic properties of Radon-Nikodym derivative,

logdµ Tn

dµ= log

dν Tn

dν+ f − f Tn

Let r ∈ e(T, log dµT

dµ

). Let E ∈ B with µ (E) > 0 and let ε > 0. Since the space

is standard Borel we are able to apply Lusin's theorem to get some E0 ⊂ E withµ (E0) > 0, such that |f |E0 −c| < ε/2. By the assumption that r ∈ e (T, µ)

there exists F ⊂ E0 with µ (F ) > 0 and n such that∣∣∣log dµTn

dµ |F −r∣∣∣ < ε/2.

Now by denition, F and n are the right choice for log dνTndν , since by F ⊂ E0

we see that ∣∣∣∣logdν Tn

dν− log

dν Tn

dν

∣∣∣∣ = |f |F −f |F Tn| < ε

hence∣∣log dνTn

dν − r∣∣ < ε and we see that r ∈ e

(T, log dνT

dν

).

Remark. We showed in the proof that the dierence of log dµTdµ and log dνT

dν isof the form f − f T for a Borel function f and, this leads us to the conclusionthat these two cocycles have the same ratio set. This can be generalized toall T -cocycles: We call a cocycle coboundary if it is of the form f − f Tnfor some Borel function f . Two T -cocycles are called cohomologous if theydier by a coboundary. The cohomology of T , then, consists on the T -cocyclesclasses modulo coboundaries and as we mentioned the ratio set is a cohomologyinvariant. For a concise representation see [Sch].

The above proposition shows that the following typology depends only on theequivalent class of µ but not on µ itself:

Denition (Krieger's Classication). Let T be a non-singular transformationof a standard Borel space (X,B, µ). We say that T is of type II if there existsan a.c.i.m. with respect to µ, that is, there exists a sigma-nite Borel measurewhich is both T -invariant and equivalent to µ. Otherwise we say that T is oftype III.We further classify type II into type II1 and type II∞ where the a.c.i.m. isnite or innite, respectively.Also, we further classify type III into type III0 if e (T, log T ′) = ∅; type IIIλfor some 0 < λ < 1 if e (T, log T ′) = |log λ|Z; and type III1 if e (T, log T ′) = R.

Proposition. If T is ergodic, then T is conservative if and only if 0 ∈ e (T, log T ′).

Corollary. Since e (T, log T ′) is closed under addition, it follows that 0 ∈e (T ) ⇐⇒ e (T, log T ′) 6= ∅. Hence all the conservative and ergodic non-singular transformations of type III are classied by their ratio set as one of thetypes IIIλ, 0 < λ ≤ 1.

20

3 ORBITAL EQUIVALENCE AND THE RATIO SET 3.1 The AssociatedFlow

Proof. Suppose T is conservative. By Maharam's theorem T is conservative.Let E ∈ B with µ (E) > 0 and ε > 0. By the conservativity of T , there existsn ≥ 1 such that

µ(E × (−ε/2, ε/2) ∩ T−n (E × (−ε/2, ε/2))

)> 0

where dµ (x, t) = dµ (x) ⊗ exp (t) dt is the invariant measure of the Maharamextension. Notice that

E × (−ε/2, ε/2) ∩ T−n (E × (−ε/2, ε/2))

⊂ T−nE × (−ε, ε) ∩ E ×∣∣log (Tn)

′∣∣ < ε

so we get thatµ(E ∩ T−nE ∩

∣∣log (Tn)′∣∣ < ε

)> 0

This shows that 0 ∈ e (T, log T ′).Suppose now that 0 ∈ e (T, log T ′). Let E ∈ B with µ (E) > 0 and ε > 0. Bythe assumption there exists n ≥ 1 such that

µ(E ∩ T−nE ∩

∣∣log (Tn)′∣∣ < ε

)> 0

and in particular µ (E ∩ T−nE) > 0. This means that no wandering set canexist for T , so T is conservative.

3.1 The Associated Flow

The construction of the associated ow was rst introduced by Krieger.This is a way to prove the full classication of orbit-equivalence typesaccording to the ratio set. We introduce the general notion of theassociated ow, showing that this is indeed an orbit-equivalence in-variant and, we prove the classication of type III1 transformationsaccording to the corresponding ratio set. For a full treatment of theKrieger's classication we refer to [HOO] and [KW].

Theorem (Ergodic Decomposition Theorem. See [GS, Theorem 1.1] for prob-ability measures and [Sch, Corollary 6.9] for sigma-nite measures.). Let T bea non-singular transformation of a standard Borel space (X,B, µ) where µ issigma-nite. Then there exists a standard Borel space (E, E , ε) and a collectionµE | E ∈ E of sigma-nite measures on (X,B), such that the following hold:

1. For every B ∈ B, the map E 7→ µE (B) is a Borel map E→ [0,∞].

2. For every B ∈ B,µ (B) =

E

µE (B) dε (E)

3. For every E ∈ E, T is ergodic and non-singular with respect to µE.

21


4. For every E,E′ ∈ E, if E 6= E′ then µE and µE′ are mutually singular.

5. For every B ∈ B invariant to T , denote Φ (B) := E ∈ E | µE (B) > 0.Then Φ (B) is a Borel set and moreover,

E = Φ (B) | B ∈ B is invariant to T

up to ε-null sets.

The space (E, E , ε) is unique up to isomorphism, and is called the space of

ergodic components of (X,B, µ) with respect to T .

Remark. Since Suppµ =⋃E∈E SuppµE we see that for a.e. x ∈ X there exists

E ∈ E such that x ∈ SuppµE . This E is unique for a.e. x ∈ X by property 4.Let π : X → E be this correspondance and by property 5 π is a borel map. Wecall π (x) the ergodic component of x.In the contexts of non-singular transformations, it can be seen that the er-godic decomposition respects the Radon-Nykodim derivative in the sense thatdµTdµ (x) = dµET

dµE(x) for µE-a.e. x ∈ X.

For this note that standard argument extends property 2 to

X

f (x) · g π (x) dµ (x) =

E

g (E) ·

X

f (x) dµE (x)

dε (E)

for every f ∈ L1 (X,µ) and g ∈ L∞ (E, ε). Now using the additional fact thatπ is a factor map w.r.t. to the identity transformation of (E, E , ε), calculationshows that

E

g (E)

X

(dµ Tdµ

(x) · f (x)

)dµE (x)

dε (E)

=

E

g (E) ·

X

dµE TdµE

(x) · f (x) dµE (x)

dε (E)

for every f and g as before. For details see [Aar, 1.0.8 and 1.0.11]. Thus for a.e.E ∈ E w.r.t. ε, we get

dµ Tdµ

(x) =dµE TdµE

(x)

for a.e. x ∈ X w.r.t. µE .

An R-action is called ow. Let T be a non-singular transformation of a stan-dard Borel space (X,B, µ). Let T (x, t) = (Tx, t− log T ′ (x)) be its Maharam

extension acting on the standard Borel space(X, B, ν

), where X = X ×R and

ν = µ⊗ e, de (t) = exp (t) dt.

22


Denition. Consider the ow (Sr)r∈R on X dened by

Sr (x, t) := (x, t+ r)

Note that the action of (Sr)r∈R is Borel and non-singular with respect to ν.

Also T commutes with Sr for every r ∈ R.Let (E, E , ε) be the space of ergodic components of

(X, B, ν

)with respect to T .

Note that the T -invariance of the the ergodic components and the commutingT Sr = SrT imply that for an ergodic component E ∈ E we have TE = Eand T SrE = SrTE = SrE so SrE is a T -invariant set. Thus the action of

(Sr)r∈R on X descends a well-dened action(Sr

)r∈R

on the space of the ergodic

components (E, E , ε). This action is also Borel and non-singular with respect to

ν. The action(Sr

)r∈R

is called the associated ow of T .

Lemma (Ergodicity of the associated ow). Consider the Z × R-action on Xdened by the joint action (n, r) 7→ TnSr. The following are equivalent:

1. The joint action is ergodic.

2. T is ergodic.

3. The associated ow is ergodic.

Proof. Here we write f = g for f, g in L∞, if the set x | f (x) 6= g (x) is anull set. We use the ergodic theorem characterization for ergodicity, that everyinvariant L∞-function is constant in the above sense.1 implies 2: Let f : X → R be a function invariant to T . Consider the functionf : X → R dened by f (x, t) = f (x) and we see that

f(T Sr (x, t)

)= f (Tx, t+ r − log T ′ (x)) = f (Tx) = f (x) = f (x, t)

Thus by the ergodicity of the joint action f is constant hence f is constant.2 implies 1: Let f : X → R be a function invariant to the joint action. Inparticular f (x, t) = f (x, t+ r) for every r ∈ R. In the case that f is continuousit follows immediately that f (x, t) = f (x) depends only on the rst coordinate.In the general case, we x an arbitrary Borel set B of nite measure and byLuzin's theorem for every ε > 0 we have f (x, t) = f (x) a.e. on B except for aset of measure at most ε. It follows that f (x, t) = f (x) with the same constantf (x) for a.e. x ∈ B. Since B is arbitrary Borel set of nite measure and sincethe space is sigma-nite, we see that f (x, t) = f (x) depends only on the rstcoordinate. Now we get by the same calculation as above one can see that fis invariant to T . By the ergodicity of T we see that f is constant hence f isconstant.1 implies 3: Let f : E → R be a function invariant to the associated ow(Sr

)r∈R

. Denote by [x, t] ∈ E the ergodic component of an element (x, t) ∈ X

23


and consider the function f : X → R dened by f (x, t) = f ([x, t]). Note thattaking element to its ergodic component is a Borel map as we mentioned before.Since T preserves the ergodic components we see that

f(T Sr (x, t)

)= f

([T Sr (x, t)

])= f ([Sr (x, t)])

= f(Sr [x, t]

)= f ([x, t]) = f (x, t)

Thus by the ergodicity of the joint action f is constant hence f is constant.3 implies 1: Let f : X → R be a function invariant to the joint action. Denef : E→ R by f ([x, t]) = f (x, t). Note that since f is invariant to T it is constanton each ergodic components, so f is well dened. By the same calculation asabove one can see that f is invariant to the joint action. By the ergodicity ofthe joint action f is constant hence f is constant.

Denition. Non-singular ows (Sr)r∈R , (Ur)r∈R over standard Borel spaces(X,B, µ) , (Y, C, ν) respectively are said to be isomorphic, if there exists a Borelbijection ϕ : X → Y such that ϕ∗µ ∼ ν and ϕSr (x) = Urϕ (x) for a.e. x ∈ X.

Theorem. [HOO, Theorem 1] The isomorphism type of the associated ow isan invariant of orbit-equivalence.

Corollary. Since orbital-equivalence type depends only on the equivalence-classof the measure, it follows that the associated ow depends only on the equivalence-class of the measure.

Proof. Let T,U be two non-singular transformations of a standard Borel spaces(X,B, µ) , (Y, C, ν), respectively. Let ϕ : X → Y be an orbit-equivalence of T,U .Dene ϕ : X → Y by

ϕ (x, t) :=

(ϕ (x) , t− log

ν ϕµ

(x)

)Then ϕ is a bijection which commutes with the ows (Sr)r∈R on X and Y . In

order to show that it commutes with the associated ows of X and Y , we leftto show that if f : Y → R is an U -invariant function then f ϕ : X → R is anT -invariant function.Denote U := ϕTϕ−1. First, note that since ϕ is an orbital-equivalence, U is in

the full group [U ]. Thus if f : Y → R is U -invariant then it is also U -invariant.

Second, we claim that ϕ T = U ϕ: By the denitions of T , U and ϕ, we have

ϕ(T (x, t)

)=

(ϕ (Tx) , t− log

dµ Tdµ

(x)− logdν ϕdµ

(Tx)

)and

U (ϕ (x, t)) =

(Uϕ (x) , t− log

dν ϕdµ

(x)− logdν Udν

(ϕ (x))

)

24


Since Uϕ = ϕT the rst coordinates are equal. As for the second coordinate,note that

dµ Tdµ

(x) · dν ϕdµ

(Tx) =dµ Tdµ

(x) · dν ϕTdµ T

(x) =dν ϕTdµ

(x)

=dν Uϕdµ

(x) =dν ϕdµ

(x) · dν Uϕdν ϕ

(x)

=dν ϕdµ

(x) · dν Udν

(ϕ (x))

Taking log and rearranging terms imply that the second coordinates are equal.These two properties of U show that

f ϕ T = f U ϕ = f ϕ

for every U -invariant function f , and the result follows.

Theorem (Krieger and Hamachi-Oka-Osikawa [HOO, Theorem 3]). Let T be anon-singular ergodic transformation of a standard Borel space (X,B, µ). Then,

1. T is of type II if and only if its associated ow is translation on R, thatis t 7→ t+ r. In this case e (T, log T ′) = 0.

2. T is of type IIIλ for 0 < λ < 1 if and only if its associated ow is t 7→ t+rmod |log λ| on the interval [0, |log λ|).

3. T is of type III1 if and only if its associated ow is the constant ow on asingleton, that is, E is a singleton and T is ergodic.

4. T is of type III0 if and only if its associated ow is free.

We prove only part 3 of this theorem as this is a fundamental tool in our work.

Denition. Let T be a non-singular transformation of a standard Borel space(X,B, µ). Let ϕ be a cocycle for T . Denote by Iϕ the sigma-algebra of BorelTϕ-invariant sets. We dene the period set to be

P (T, ϕ) :=r ∈ R | SrE = E for every E ∈ Iϕ

Theorem (Schmidt [Sch, Theorem 5.2]). For ergodic T ,

e (T, ϕ) = P (T, ϕ)

Proof. Let ν = µ⊗m be the measure on the skew-product. We show rst thate (T, ϕ) ⊃ P (T, ϕ). As the reader shell see we will not need T to be ergodic toshow that. Assume r /∈ e (T, ϕ), then there exists E with µ (E) > 0 and ε > 0,such that for every n ∈ Z,

µ(E ∩ T−nE ∩ |ϕn − r| < ε

)= 0

25


LetE :=

⋃n∈Z

Tnϕ (E × (−ε/2, ε/2))

Then E ∈ Iϕ with ν(E)> 0. Hence if r ∈ P (T, ϕ) we would have

E = SrE =⋃n∈Z

Tnϕ (E × (r − ε/2, r + ε/2))

But by the assumption and the denition of ν we see that ν(E ∩ SrE

)= 0,

hence r /∈ P (T, ϕ).We show now that e (T, ϕ) ⊂ P (T, ϕ). Assume r /∈ P (T, ϕ), then there exists

E ∈ Iϕ with ν(E)> 0 such that E ∩ SrE = ∅. For every x ∈ X consider the

berEx :=

s ∈ R | (x, s) ∈ E

Note that (Tx, s) = T (x, s+ ϕ (x)) so by T -invariance of E we see that (Tx, s) ∈E ⇐⇒ (x, s+ ϕ (x)) ∈ E thus ETx = Ex + ϕ (x). It follows that the Borelmap x 7→ m (Ex) is T -invariant hence constant by ergodicity of T . In particularm (Ex) > 0 for a.e. x ∈ X.Let B0 be the space

B0 := B ∈ BR | 0 < µ (B) <∞ ⊂ BR

Endow B0 with the metric

η (B1, B2) := µ (B14B2)

Then (B0, η) is a complete separable metric space (See [Bo, 1.12(iii), p. 53]).Also the maps B0 × B0 → B0 of union (B1, B2) 7→ B1 ∪ B2 and intersection(B1, B2) 7→ B1∩B2 are continuous with respect to η and also the map B0×R→B0 of translation (B, r) 7→ B + r is continuous with respect to η.For a.e. x ∈ X we have m (Ex) > 0 so also m (Ex ∩ [−n, n]) > 0 for some n ∈ N.Write n (x) = inf n ∈ N | m (Ex ∩ [−n, n]) > 0 and dene

Θ (x) := Ex ∩ [−n (x) ,n (x)]

Then Θ : X → B0 is a Borel map. By Luzin's theorem there exists c > 0and a compact set Y0 ⊂ X with µ (Y0) > 0, such that Θ |Y0

is continuous andm (Θ (y)) ≥ c > 0 for every y ∈ Y0.Let Φ : Y0 × Y0 × R→ R be the map

Φ (y, y′, t) := m (Θ (y) ∩ (Θ (y′) + t))

Then Φ is continuous as a composition of translation and intersection on themap E 7→ m (E). Also we have by denition Φ (y, y, 0) = m (Θ (y)) ≥ c for everyy ∈ Y0. It follows that again there exists a neighborhood Y × Y × (−ε, ε) where

26


Y ⊂ Y0 is a compact set with µ (Y ) > 0 and ε > 0 such that Φ (y, y′, t) ≥ c/2for every y ∈ Y and |t| < ε.Let n ∈ Z and x ∈ T−nY . Notice that for t ∈ Θ (Tnx)+r we have (Tnx, t− r) ∈E so (Tnx, t) = Sr (Tnx, t− r) ∈ SrE. On the other hand for t ∈ Θ (x)+ϕn (x)we have (x, t− ϕn (x)) ∈ E and by T -invariance we see that (Tnx, t) ∈ E. Bythe assumption E ∩ SrE = ∅ we conclude that for every n ∈ Z and x ∈ T−nY ,

(Θ (Tnx) + r) ∩ (Θ (x) + ϕn (x)) = ∅

This implies that

Φ (x, Tnx, ϕn (x)− r) = m (Θ (x) ∩ (Θ (Tnx) + ϕn (x)− r)) = 0

and by the denition of Φ on Y ×Y ×(−ε, ε) we see that ϕn (x)−r /∈ (−ε, ε).

Corollary. [Sch, Corollary 5.4] Tϕ is ergodic ⇐⇒ T is ergodic and P (T, ϕ) =R ⇐⇒ T is ergodic and e (T, ϕ) = R . In particular, applying this to theRadon-Nikodym cocycle of the half stationary shift T , we get that the Maharamextension T is ergodic if and only if T is ergodic and of type III1.

Lemma. Let B1, B2 be Borel sets of R with Lebesgue measure m (Bi) > 0 fori = 1, 2. Then there exists an open interval ∅ 6= I ⊂ R such that for every t ∈ I,

m (B1 ∩ (B2 + t)) > 0

Proof. We assume WLOG 0 < µ (Bi) < ∞ for i = 1, 2. Let f : R → R be thefunction f (t) = m (B1 ∩ (B2 + t)). Then f is non-negative and is continuousby the continuity of intersection and translation we mentioned above. Thus it isenough to show that m (f > 0) > 0. This follows from Fubini's theorem whichshows that ‖f‖1 = m (B1) ·m (B2) > 0.

Proof. (Corollary). The second equivalence in the corollary follows from e (T, ϕ) =P (T, ϕ). We show the rst equivalence. On one hand, if Tϕ is ergodic then Iϕ =∅, X × R so P (T, ϕ) = R. Assume now that T is ergodic and P (T, ϕ) = R.Let E ∈ Iϕ and Ex :=

s ∈ R | (x, s) ∈ E

as before. Since E ∈ Iϕ we see

that the map x 7→ m (Ex) is T -invariant so by ergodicity of T it is constanta.e. on X. Since P (T, ϕ) = R we have that Ex = Ex + r for every r ∈ R anda.e. x ∈ X, so we can x a dense countable set rn | n ∈ N ⊂ R such thatm (Ex4 (Ex + rn)) = 0 for a.e. x ∈ X and every n ∈ N.For some x ∈ X for which the above holds, assume by contradiction that

m(Ex

)> 0. Then by the lemma there exists an open interval Ix ⊂ R such that

m(

(Ex + t) ∩ Ex

)> 0 for every t ∈ Ix. But for some rn ∈ Ix we must have

m(

(Ex + rn) ∩ Ex

)= m

(Ex ∩ E

x

)= 0, a contradiction. Thus m

(Ex

)= 0

and since x is arbitrary from some full measure set we see that m(Ex

)= 0 a.e.

on X. By the denition of ν and Fubini's theorem we conclude that ν(E)

= 0

and Tϕ is ergodic.

27

4 THE ORBITAL EQUIVALENCE RELATION

4 The Orbital Equivalence Relation

For a concise treatment of orbit theory from perspective of equivalence relationssee [KM].Let (X,B) be a standard Borel space. A Borel equivalence relation on Xis a Borel set R ∈ B ⊗ B such that the relation x ∼ y ⇐⇒ (x, y) ∈ R on Xis an equivalence relation. For an equivalence relation R and a set A ∈ B, thesaturation of A is the set

R (A) :=⋃x∈XR (x) , where R (x) = y ∈ X | (x, y) ∈ R

A Borel equivalence relation R is nite (resp. countable) if for all x ∈ X,R (x) is nite (resp. countable) and we say that R is hypernite if thereexists an increasing sequence of nite equivalence relation R1 ⊂ R2 ⊂ · · · ⊂ Rsuch that R =

⋃∞n=1Rn.

Two Borel equivalence relationsR,R′ over standard Borel spaces (X,B) , (X ′,B′)are isomorphic, if there exists a Borel bijection ϕ : X → X ′ such that

(x, y) ∈ R ⇐⇒ (ϕ (x) , ϕ (y)) ∈ R′

A Borel equivalence relation R is non-singular with respect to a measure ν on(X,B), if for every Borel set A,

ν (R (A)) = 0 ⇐⇒ ν (A) = 0

and is ergodic with respect to a measure ν on (X,B), if for every Borel set A,

R (A) = A =⇒ ν (A) = 0 or ν(A)

= 0

Theorem (Feldman-Moore [FM]; Weiss [We]). Every Borel countable equiv-alence relation R admits a countable group Γ acting on (X,B, ν) such thatR = RΓ; That is,

(x, y) ∈ R ⇐⇒ ∃γ ∈ Γ, γx = y

This group is not necessarily unique. Furthermore, R is hypernite if and onlyif it is generated by a single transformation; that is, R admits a Borel Z-actionfor which R = RZ.

The notion of invariance of a measure\set\function with respect to R, coincideswith notion of invariance with respect to all the elements of a correspondinggroup Γ, independently on the choice of Γ. Also there is an analogue for theergodic decomposition for a countable Borel equivalence relation. See chapter Iin [KM].

Denition. Let T be a non-singular transformation of a standard Borel space(X,B, µ). We dene the corresponding orbital equivalence relation onX×Xby

OT := (x, y) ∈ X ×X | ∃n,m ∈ N, Tnx = Tmy

28


and the tail equivalence relation on X ×X by,

TT := (x, y) ∈ X ×X | ∃n ∈ N, Tnx = Tny

Note that for a measurable set A, the saturation with respect to the tail equiv-alence relation is

TT (A) =⋃x∈ATT (x) =

⋃n∈N

T−n (TnA)

since in general Tnx = Tny ⇐⇒ y ∈ T−n (Tnx).

Example (The one-sided shift). Consider the one-sided shift σ : SN → SNwhere S is a nite set. The tail equivalence relation Tσ is indeed hypernite sinceit is the countable union of the nite relations RN , N ∈ N, where (x, y) ∈ RNif and only if xn = yn for every n > N . One can see that ΓTσ can be chosento be the group of all automorphisms SN → SN that perform changing of thesymbols of S on nitely many coordinates. As we mentioned in Feldman-Mooretheorem, a group corresponding to a hypernite Borel equivalence relation isnot unique in general. This example demonstrates this fact; we show in thenext example that the Z-action induced by the odometer also describes ΓTσ .

Example (The non-singular odometer). Equip SN with a structure of abeliangroup as follows: Identify S = 0, 1, . . . , d− 1 for some d ∈ N. First notice thatevery non-negative integer N can be written in a unique way as N =

∑∞i=0 xid

εi ,where xi ∈ 0, 1, . . . , d− 1 and εi ∈ 0, 1, so N can be represented by the se-quence (x0, x1, . . . , xk, 0, 0, . . . ) ∈ SN for some k. We dene the commutativesum of such two representations corresponding to some non-negative integersN,M to be the corresponding representation of N +M . This rule has an induc-tive description which can be extended in a natural way to dene a commutativesum of any two elements of SN, not only these with nitely many non-zero co-ordinates. This extended denition of adding operation for all elements of SNhas an additive inverse with respect to the zero element (0, 0, 0, . . . ), as one cansee for instance that

− (1, 0, 0, . . . ) = (d− 1, d− 1, d− 1, . . . )

and in an inductive way we can nd an additive inverse to any element of SN.Thus SN is an abelian group with respect to this addition.Now we dene the odometer τ : SN → SN to be the invertible transformation

τ : x 7→ x+ (1, 0, 0, 0, . . . )

In order to understand the measure theoretic properties of the odometer, we con-

sider the monotone class of cylinders of the form C(j)k := [d− 1, . . . , d− 1, j]

k1 for

k ∈ N and j ∈ S. The action of τ on this cylinders is τC(j)k = [0, . . . , 0, j + 1]

k1 ,

as this is simply adding 1 to the telescoping equality

(d− 1)(1 + d+ · · ·+ dk−1

)+ jdk = dk − 1 + jdk = (j + 1) dk − 1

29


If we consider any product measure µ =∏n∈Z µn with some δ > 0 such that

µn (j) ≥ δ > 0 for every n ∈ N and j ∈ S, then the odometer is non-singulartransformation with respect to µ. This can be seen by calculation of Radon-

Nikodym derivative τ ′ = dµτdµ on the monotone class of cylinders C

(j)k which

yields the formula

τ ′ (x) =

φ(x)∏i=1

µi ((τx)i)

µi (xi)

whereφ (x) := min k ∈ N | xk 6= d− 1

Consider the orbital equivalence relation of the odometer Oτ . We claim thatthe corresponding group Γ = ΓOτ can be chosen to be the group of automor-phisms SN → SN that perform permutation of the symbols of S on nitely manycoordinates. This follows from The odometer property (See for example [Aar,"The Adding Machine", Chapter 1.2]) which says that for every x ∈ SN and apositive integer N ,((

τkx)

1, . . . ,

(τkx

)N

)| 0 ≤ k ≤ dN − 1

= 0, 1, . . . , d− 1N

Thus we can see thatTσ = Oτ

Since τ is invertible we conclude that the odometer is the tail-action of theone-sided shift.The following proposition together with the identication Tσ = Oτ , show thatif µ is any Borel measure on SN for which the one-sided shift is exact, then theodometer is ergodic with respect to µ.

Proposition (Hawkins [Haw]). Let T be a non-singular transformation of astandard Borel space (X,B, µ).Then:

1. T is non-singular if and only if OT is non-singular.

2. T is ergodic if and only if OT is ergodic.

3. If T is non-singular then TT is non-singular.

4. Recall that a transformation S is called exact if⋂∞n=1 T

−nB is trivial.Then T is exact if and only if TT is ergodic.

Proof.

1. If T is singular then there exists A with µ (A) > 0 and µ(T−1A

)= 0.

But A ⊂ OT(T−1A

)so 0 < µ (A) ≤ µ

(OT

(T−1A

)), so OT is singular.

If OT is singular then there exists A with µ (A) = 0 and µ (OT (A)) > 0.Hence there exists n such that µ (TnA) > 0, so T is singular.

30


2. Notice that the result follows from the observation that for every Borelset A,

T−1A = A ⇐⇒ OT (A) = A

Since OT (x) = T−1OT (x) we get that

OT (A) = OT(T−1A

)= T−1OT (A)

and the result follows.

3. This is straightforward from TT (A) =⋃n∈N T

−n (TnA).

4. We will show that∞⋂n=1

T−nB = A ∈ B | TT (A) = A

and this would be enough, since exactness of T means that the left side istrivial mod µ and ergodicity of TT means that the right side is trivialmod µ.

We have the following correspondence: On one hand, if A ∈⋂∞n=1 T

−nBthen A = T−n (TnA) for every n ≥ 1. Thus for every such A,

TT (A) =

∞⋃n=1

T−n (TnA) = A

On the other hand, for every Borel setA withA = TT (A) =⋃n∈N T

−n (TnA)we clearly have that A ∈

⋂∞n=1 T

−nB.

Corollary. By the facts we mentioned before we see the follows: An equiv-alence relation R is hypernite if and only if it is orbital-equivalent to somenon-singular and invertible transformation V . Therefore for a non-singulartransformation T , the equivalence relation R := TT is hypernite if and only if

TT = OVwhere V is a non-singular and invertible transformation. We call such V thetail action of T . The exactness of T means that TT is trivial and ergodicityof V means that OV is trivial, so it follows that T is exact if and only if thecorresponding tail action V is ergodic.

The following constructions are due to [ANS].

Denition. Let R be a Borel equivalence relation over a standard Borel space(X,B, µ). We say that ψ : R → R is an R-orbital cocycle, if for every(x, y) , (y, z) ∈ R,

ψ (x, y) + ψ (y, z) = ψ (x, z)

For R and R-cocycle ψ, we dene the skew-product equivalence relation overX × R to be

Rψ := ((x, t) , (y, s)) | (x, y) ∈ R , t− s = ψ (x, y)

31


Denition. Let T be a non-singular transformation of (X,B, µ). Given anyfunction ϕ : X → R, denote for N ≥ 1

ϕN := ϕ+ ϕ T + · · ·+ ϕ TN

Dene a TT -cocycle ϕ : TT → R for (x, y) ∈ TT with TNx = TNy by

ϕ (x, y) = ϕN (x)− ϕN (y)

and a OT -cocycle ϕ : OT → R for (x, y) ∈ OT with TNx = TMy by

ϕ (x, y) = ϕN (x)− ϕM (y)

Remark. Let ϕ : X → R and consider the skew-product Tϕ (x, t) = (Tx, t− ϕ (x)).Then we have

TTϕ = (TT )ϕ

In particular, if V is a tail action for T , that is TT = OV , then

TTϕ = (OV )ϕ

where the right equivalence relation gives the condition for ((x, t) , (y, s)) ∈(OV )ϕ where (x, y) ∈ OV with y = V Nx, N ∈ Z (since V is invertible) by

t− s = ϕ (x, y) = ϕ(x, V Nx

)Consider the skew-product of V and the orbital cocycle generated by the func-tion ψ (x) := ϕ (x, V x), we see that

TTϕ = OVψ

This gives the following insight:

Tϕ is exact ⇐⇒ T Tϕ = OVψ is ergodic ⇐⇒ Vψ is ergodic

Denition. For an equivalence relation R with a corresponding group Γ, wedene the Radon-Nikodym cocycle of R to be the orbital cocycle

ψ (x, γx) := logdµ γdµ

(x)

for every x ∈ X and γ ∈ Γ. This is well dened almost everywhere independentlyon the choice of Γ, since for every two Borel automorphisms γ, θ with γx = θxwe have that dµγ

dµ = dµθdµ a.e. on E := γ = θ.

32

5 KAKUTANI'S CRITERION

5 Kakutani's Criterion

Kakutani's criterion [Kak] is a fundamental quantitative test forequivalence of product measures and Markov measures. In particularit is also a quantitative test for non-singularity of a transformation.The main and more general result is called Kakutani's dichotomy.We introduce here Kakutani's dichotomy as appears in [Shi, Section6 in Chapter VII] and use it to establish Kakutani's criterion forproduct measures and in the next section it will be used also forMarkovian measures.

Every sequence of independent random variables X := (X1, X2, . . . ) over aprobability space (Ω,F , P ) induces a probability product measure P on the Borelspace

(RN,B

)by P (E) := P (X ∈ E). Denote the probability measure on the

n-th coordinate of(RN,B

)by Pn (E) := P (Xn ∈ E). That is, P =

∏n∈N Pn.

Assume we have two such sequences X (i), i = 1, 2 with their correspondingprobability measures P(i), i = 1, 2 on

(RN,B

). Suppose that P(1) is locally

absolute continuous with respect to P(2), that is, for every n we have P(1)n

P(2)n . What does it say about the possibly absolute continuity of P(1) with

respect to P(2)?

Theorem (Kakutani's dichotomy). In the situation as above, either P(1) P(2)

or P(1) ⊥ P(2).Moreover, we have an explicit test identies what possibility occurs: Denote

mn :=dP(1)n

dP(2)n

and Mn := mn ·m−1n−1 · 1mn−1 6=0. Let Bn be the Borel sub-sigma-

algebra of RN supported on the coordinates 1, . . . , n. Then the dichotomy isdetermined by the 0− 1 event

B :=

∑n≥1

(1−E

[√Mn | Bn−1

])<∞

as P(1) (B) = 1 ⇐⇒ P(1) P(2) and P(1) (B) = 0 ⇐⇒ P(1) ⊥ P(2).

Remark. As one can notice, the process (mn)∞n=1 with respect to the natural

ltration Fn := σ (X1, . . . , Xn), is a martingale which is bounded as E [mn] =1 for all n ∈ N. Hence by the the martingale convergence theorem m∞ :=limn→∞mn exists. In the proof as in [Shi, Section 6 in Chapter VII] it is beingshown that B = m∞ <∞.

Proposition (Kakutani's criterion). Let µ =∏n∈N µn, ν

∏n∈N νn be a pair of

product measures on X = SN, where S is a nite set. Suppose µ |Bn , ν |Bn areequivalent for every n. Then µ, ν are equivalent if and only if∑

n≥0

∑s∈S

(√µn (s)−

√νn (s)

)2

<∞

33


Proof. Notice that the independence of the coordinates yields that Mn is mea-surable with respect to the n-th coordinate alone, namely Mn is independenton Bn−1. Hence

E[√

Mn | Bn−1

]= E

[√Mn

]=∑s∈S

µn (s)

√νn (s)

µn (s)=∑s∈S

√µn (s) νn (s)

which shows that

2 ·(

1−E[√

Mn | Bn−1

])= 1 + 1− 2

∑s∈S

√µn (s) νn (s)

=∑s∈S

(µn (s) + νn (s)− 2

√µn (s) νn (2)

)=∑s∈S

(√µn (s)−

√νn (s)

)2

and the criterion follows from the test in Kakutani's dichotomy.

Corollary (Non-singularity of the shift). Let µ =∏n∈Z µn be a product measure

on X = SZ. Then the shift T : X → X is non-singular with respect to µ if andonly if ∑

n∈Z

∑s∈S

(√µn (s)−

√µn−1 (s)

)2

<∞

In this case, calculation of the Radon-Nikodym derivative on cylinders yields theformula

dµ TN

dµ(x) =

∏n∈Z

µn−N (xn)

µn (xn)

We further introduce here the Hellinger distance as in [Kak]:

Denition. Let µ =∏n∈N µn, ν

∏n∈N νn be a pair of product measures on

X = SN, where S is a nite set. Let λ be some measure on X such that µ λand ν λ (For example λ = µ+ ν). Denote

ψ =

√dµ

dλ, ψ′ :=

√dν

dλ

so that ψ and ψ′ are in L2 (X,B, λ) with ‖ψ‖2 = ‖ψ′‖2 = 1. Dene theHellinger distance of µ and ν by

ρ (µ, ν) = 〈ψ,ψ′〉

One can see that ρ (µ, ν) does not depend on λ and that ρ (µ, ν) = ρ (ν, µ).Note also that ρ (µ, ν) = 0 if and only if µ ⊥ ν. In any other case we have byCauchy-Schwartz inequality 0 < ρ (µ, ν) ≤ 1 where ρ (µ, ν) = 1 if and only ifµ = ν.Finally, dene the metric

d (µ, ν) := ‖ψ − ψ′‖ =√

2 · (1− ρ (µ, ν))

34


As a consequence of the general inequalities 1 − 1/x ≤ log x ≤ 1 − x for every0 < x, we have the following property:

Claim. [Kak] Denote σ (µ, ν) := − log ρ (µ, ν). There exists constant C > 1such that

C−1 · σ (µ, ν) ≤ d (µ, ν)2 ≤ C · σ (µ, ν)

whenever σ (µ, ν) or d (µ, ν) are small enough.

Example. Let µ =∏n∈Z µn be a product measure on X = SZ. Assume that

the shift T : X → X is non-singular with respect to µ. Note that using the

formula(TN)′

(x) =∏n∈Z

µn−N (xn)µn(xn) we get

ρ(µ, µ T k

)=∑n∈Z

∑s∈S

√µn−N (s)

µn (s)· µn (s) =

∑n∈Z

∑s∈S

(√µn−N (s)µn (s)

)so

d(µ, µ TN

)2= 2 ·

(1− ρ

(µ, µ TN

))=∑n∈Z∑s∈S

(√µn−N (s)−

√µn (s)

)2

This is indeed nite for every N ≥ 1, as for N = 1 this is Kakutani's criterionfor non-singularity and for N ≥ 1 we have inductively(√

µn−N (s)−√µn (s)

)2

=(√

µn−N (s)−√µn−(N−1) (s)

)2

+(√

µn−(N−1) (s)−√µn (s)

)2

+ 2 ·(√

µn−N (s)−√µn−(N−1) (s)

)·(√

µn−(N−1) (s)−√µn (s)

)and by Cauchy-Schwartz inequality, the series over the right hand side converges.

Proposition. [Kos, Lemma 2.2] If ρ(µ, µ TN

)< ∞ for every N ≥ 1 and∑∞

N=1 ρ(µ, µ TN

)<∞, then T is dissipative.

Proof. We show that∑∞N=1

(TN)′<∞ a.e. on X so by Hopf's decomposition

theorem T is dissipative. We have that

ρ(µ, µ TN

)=

X

√(TN )

′dµ ≥

|(TN )′|>1

√(TN )

′dµ ≥ µ

(∣∣∣(TN)′∣∣∣ > 1)

so by the assumption∑∞N=1 µ

(∣∣∣(TN)′∣∣∣ > 1)<∞ and by Borel-Cantelli lemma

for a.e. x ∈ X there exists N0 = N0 (x) such that(TN)′

(x) ≤√

(TN )′(x) ≤ 1

for every N ≥ N0. The above inequality also shows that∑∞N=1

√(TN )

′< ∞,

so we conclude that∑∞N=1

(TN)′<∞.

35

6 FUTURE RESEARCH: INHOMOGENEOUS MARKOV SUBSHIFT

6 Future Research: Inhomogeneous Markov Sub-

shift

Here we introduce the notion of inhomogeneous Markov subshift as a general-ization of the classical Bernoulli shift. Our main theorem will be established forBernoulli shift, but we expect to generalize it in the near future to inhomoge-neous Markov subshift, using a similar method of proof.Consider the space SZ for some nite set S, |S| = d ∈ N. Let A be a d × dmatrix with entries in 0, 1. Consider the subspace

ΣA :=

(xn)n∈Z ∈ SZ | A (xnxn+1) = 1 ∀n ∈ Z

One may think of A as an adjacency matrix of the directed graph (S, E) whereE = (x, y) ∈ S × S | A (x, y) = 1. Then the space ΣA consists on all bi-innitetrajectories along this directed graph. The elements of SZ which are also in ΣAare called A-admissible. In the same way, a cylinder of SZ consists only onA-admissible elements is called A-admissible.Note that we have the usual shift transformation T : ΣA → ΣA. The Borelspace (ΣA,B, T ), where B is the sigma-algebra generated by the A-admissiblecylinders, is called subshift (of nite type).As in classical Markov chains, a stochastic d×d matrix P is called irreducibleif for every s, t ∈ S with A (s, t) > 0, there exists n ∈ N such that Pn (s, t) > 0.This means that the Markov chain is ergodic with respect to the shift. Also Pis called aperiodic if for every s ∈ S we have gcd n | Pn (s, s) > 0 = 1.It is well-known that P is irreducible and aperiodic if and only if there existsN ∈ N such that all the entries of PN are positive.

Denition (Inhomogeneous Markov Subshift). Let ΣA ⊂ SZ be a subshift.Fix a sequence πn, Pn | n ∈ Z where for every n ∈ Z, πn is a distributionon S and Pn is a stochastic d × d matrix with A (s, t) > 0 ⇐⇒ Pn (s, t) > 0.Assume that πnPn (s) = πn+1 (s) for every n ∈ Z and s ∈ S. Then by theKolmogorov's extension theorem there exists a Borel measure µ dened on A-admissible cylinders by

µ(

[~a]n+mn

):= πn (an) · Pn (an, an+1) · · · · · Pn+m−1 (an+m−1, an+m)

If we consider the random variables Xn : ΣA → S dened by Xn

((xk)k∈Z

)= xn

for every n ∈ Z, then for every a1, . . . , ak, b ∈ S and n1, . . . , nk ∈ Z,

µ (Xnk+1 = b | Xnk = ak, . . . , Xn1= a1) = Pnk (ak, b) = µ (Xnk+1 = b | Xnk = ak)

so (Xn)n∈Z is a Markov chain. The shift T over the standard Borel space(ΣA,B, µ) is called inhomogeneous Markov subshift (IMS).If there exists N ∈ N such that all the entries of AN are positive, then (ΣA,B, µ)is called topologically mixing (TM). In this case, every d×d stochastic matrixP with A (s, t) > 0 ⇐⇒ P (s, t) > 0 is irreducible and aperiodic.We say that µ is half-stationary if Pn = P0 and πn = π0 for every n ≤ 0.

36


Our motivation in extending our theorem to Markov subshifts comes fromMarkov partition of a toral Automorphism and especially the golden meanautomorphism. We introduce here not fully formally the notion of Markov

partition, a tool which provides a link between smooth dynamical systems andsymbolic dynamical systems (that is, a product space and the shift transforma-tion). For a more comprehensive treatment of Markov partitions see [KH, page591].One of the simplest examples is the connection between real numbers and theirrepresentations by digits: Consider the dynamical system ([0, 1] ,m, f) where mis Lebesgue measure and f (x) = bx mod 1 for some b ∈ N. Consider also the

dynamical system(0, 1, . . . , b− 1N , µ, σ

)where µ =

∏n∈N (1/b, 1/b, . . . , 1/b)

and σ is the one-sided shift. Then the base b representation map [0, 1] →0, 1, . . . , b− 1N is an isomorphism of measure preserving transformations. Withthis isomorphism, the partition of [0, 1] into I0 ∪ I1 ∪ · · · ∪ Ib−1 where Ik :=[k/b, (k + 1) /b) is a Markov partition with respect to the base b representa-tion, in the sense that the set x ∈ [0, 1] | fn (x) ∈ Ik corresponds to the setx ∈ 0, 1, . . . , b− 1N | (σnx)0 = xn = k

for every k ∈ 0, 1, . . . , b− 1 and

n ∈ N. This correspondence tells as that for a full measure set in [0, 1], eachelement x is determined by the information of which partition-set Ik containsfn (x) for every n ∈ N.In this example the corresponding shift has a structure of a product measurerather then a Markov measure. Also, in this example we get the full shift ratherthen a subshift. In a more general setting, a Markov partition is correspondingto an inhomogeneous Markov subshift.Adler and Weiss [AW] for dimension 2 and later Sinai and Bowen [Bow] for every dimension proved that for every hyperbolic automorphisms of thetorus (hyperbolic means that it has no eigenvalues of modulus 1) there existsa Markov partition. Yet, in general it is hard to nd such partition and itsstructure can be extremely complicated. Adler and Weiss [Adl] constructed aMarkov partition for the measure preserving toral automorphism f : T2 → T2

dened by

f (x, y) := (bx+ yc , x) =

(1 11 0

)(x

y

)mod 1

They showed that the corresponding Markov partition is of size 3 and it is notthe full shift but the subshift ΣA ⊂ 0, 1, 2Z dened by the matrix

A :=

1 0 11 0 10 1 0

The result is an isomorphism between

(T2,m, f

)and the topologically mixing

inhomogeneous Markov subshift (ΣA, µ, T ), where µ is the homogeneous Markov

37


measure dened by

π =

1/√

5

1/ϕ√

5

1/ϕ√

5

, P =

ϕ1+ϕ 0 1

1+ϕϕ

1+ϕ 0 11+ϕ

0 1 0

where ϕ :=

(1 +√

5)/2 is the golden mean constant.

In order to extend our proof to the setup of topologically mixing IMS, we need toadjust some of the tools we used in Bernoulli shifts. Among them is a criterionfor exactness of the one-sided Markov subshift (the following proposition 2), asin this case its natural extension, the two-sided Markov subshift, has K-property.Also we will need a criterion for the non-singularity of the shift (the followingpart 1 of proposition 3). Finally, for a half-stationary Markovian measure µdened by πn, Pn | n ∈ Z, we need a criterion whether µ is equivalent to thehomogeneous measure ν (i.e. shift invariant) dened by π0 and P0 (the followingpart 2 of proposition 3).

Proposition 1 (Koslo [Kos1, Proposition 6]). Let µ dened by πn, Pn | n ∈ N.If there exists C > 0 and N ∈ N such that (Pn · Pn+1 · · · · · Pn+N ) (s, t) ≥ C forevery s, t ∈ S and n ∈ N, then the one-sided shift

(SN, µ, σ

)is exact.

Proposition 2. Let (ΣA, µ) be a topologically mixing IMS where µ is a half-stationary measure dened by πn, Pn | n ∈ Z. Then:

1. The shift T is non-singular with respect to µ if and only if

∞∑n=1

∑s∈S

(√Pn (xn, s)−

√Pn−1 (xn, s)

)2

<∞

a.e. with respect to µ.

2. let ν be the Markov measure dened by π := π0 and P := P0. Then µ ∼ νif and only if

∞∑n=1

∑s∈S

(√P (xn, s)−

√Pn (xn, s)

)2

<∞

a.e. with respect to µ.

Since the measure ν is shift invariant, in case of µ ∼ ν we see that ν isan a.c.i.m. for (ΣA, µ, T ) and it is of type II1.

Proof. For every n ∈ N let Bn be the sigma-algebra supported on the coordinates−n,−n+ 1, . . . , 0, 1, . . . , n. We use Kakutani's dichotomy for the ltration(Bn)n∈N in both parts, so our task is to calculate a formula for E

[√Mn | Bn−1

],

where Mn = mn ·m−1n−1 and mn is the local Radon-Nikodym derivative of the

given measures on Bn. The assumption of irreducibility means that there exists

the local Radon-Nikodym derivative mn =dµT |Bndµ|Bn

given by the formula

mn (x) =

n−1∏k=0

Pk−1 (xk, xk+1)

Pk (xk, xk+1)

38


as well as the local Radon-Nikodym derivative mn =dν|Bndµ|Bn

given by the formula

mn (x) =

n−1∏k=0

P (xk, xk+1)

Pk (xk, xk+1)

That is, µT and ν are both locally equivalent to µ with respect to the ltrationBn | n ∈ N.We present the calculation for the formula of the rst part about non-singularity,

and the second part is being similar. As for mn =dµT |Bndµ|Bn

(x) we have by the

above formula that

Mn (x) = mn (x) ·m−1n−1 (x) =

Pn−2 (xn−1, xn)

Pn−1 (xn−1, xn)

thus

E[√

Mn | Bn−1

](x) =

∑t,s∈S

√Mn

([t]−n−n ∩ [~x]

n−1−(n−1) ∩ [s]

nn

)· π (t) · Pn−1 (xn−1, s)

=∑t,s∈S

√Pn−2(xn−1,s)Pn−1(xn−1,s)

· π (t) · Pn−1 (xn−1, s)

=∑s∈S

√Pn−2 (an−1, s) · Pn−1 (an−1, s)

= 1− 12

∑s∈S

(√Pn−2 (xn−1, s)−

√Pn−1 (xn−1, s)

)2

and the criterion for non-singularity of the shift follows.

39

Part II

Half-Stationary Bernoulli Shift is of

Type II1 or III1

Let X = 0, 1, . . . , d− 1Z and µ =∏n∈Z µn be a half-stationary product mea-

sure where µn = (p0, p1, . . . , pd−1) for n ≤ 0 and µn =(p

(n)0 , p

(n)1 , . . . , p

(n)d−1

)for n > 0. Let T : X → X be the shift and ν =

∏n∈Z (p0, p1, . . . , pd−1) the

stationary measure (i.e. shift invariant). Denote X+ = 0, 1, . . . , d− 1N andlet σ : X+ → X+ be the one-sided shift.Our result deals with the conservative half-stationary shift. Note that as wementioned earlier, this form of half-stationary shift has K-property with respectto the sub-sigma-algebra of the positive coordinates, hence it is either conser-vative or dissipative.

Theorem. Suppose that the shift T is non-singular and conservative with re-spect to µ. If T is not of type II1 with respect to µ, in particular µ is notequivalent to ν, then T is of type III1.

Remark. As we will mention in the proof, using Kakutani's criterion to deter-mine whether µ is equivalent to ν, we can derive a quantitative test whether Tis type II1 or III1. In particular it will be clear that there exist Bernoulli shiftsof both types.

The Structure of the proof: Our goal will be to establish that Tσlog T ′ isan ergodic equivalence relation, assuming that µ is not equivalent to ν.Then by the basic properties of the tail equivalence relation it follows thatσlog T ′ is exact transformation, so its natural extension has K-property.To identify the natural extension of σlog T ′ we note that T is the naturalextension of σ and that log T ′ is measurable with respect to the positivecoordinates, hence the Maharam extension T (= Tlog T ′) is the natural

extension of σlog T ′ and T has K-property. By Maharam's theorem T is

conservative, so we conclude by Silva-Thieullen theorem that T is ergodic.Finally, by Schmidt's theorem we see that T is of type III1.

In order to show that Tσlog T ′ is ergodic, we nd an orbital cocycle ψ forwhich

Tσlog T ′ = Oψwhere the equivalence relation O is dened by the group of transforma-tions changing nitely many coordinates. In other words, O is the orbitalequivalence relation of the odometer. By Schmidt's theorem ergodicity ofOψ is equivalent to e (O, ψ) = R. Therefore, showing that e (O, ψ) = Rwill be the main eort in our proof.

40

First notice that Tσ = O and recall that Tσlog T ′ consists on ((x, t) , (y, s)) for

which (x, y) ∈ Tσ and t − s = log(TN)′

(x) − log(TN)′

(y). Thus we dene ψby

ψ (x, y) = log(TN)′

(x)− log(TN)′

(y)

for every (x, y) ∈ O and N ≥ 1 for which σNx = σNy. We get immediately bythe denition of equivalence relation for cocycles that Tσlog T ′ = Oψ.

Writing µn (a) =∏d−1i=0

(p

(n)i

)1a=i, we see that for an O-transformation R

changing the coordinates 1, . . . , N we have the formula

ψR (x) = log(TN)′

(x)− log(TN)′

((Rx)n)

=∑Nn=1

∑d−1i=0

(1xn=i − 1(Rx)n=i

)log pi

p(n)i

Remark. For the odometer we have the formula

log τ ′ (x) =

φ(x)∑n=1

logµn ((τx)n)

µn (xn)

where φ (x) := min n ∈ N | xn 6= d− 1. We mention here that a calculationyields the formula

ψR (x) = log Φ (x) + (φ (x)− 1) logpd−1

p0+ log τ ′ (x)

where Rx = τx and

Φ (x) :=pxφ(x)pxφ(x)+1

In particular we see that when (p0, p1, . . . , pd−1) = (1/d, 1/d, . . . , 1/d) is the uni-form distribution, we get that ψR = log τ ′. It follows that in this case, our resultsays that the Maharam extension of the odometer is ergodic and the odometer,if is not of type II1, is of type III1.

In order to compute e (O, ψ) = R, we rely on the asymptotics of the sequence(p

(n)0 , p

(n)1 , . . . , p

(n)d−1

)∞n=1

and (p0, p1, . . . , pd−1), as can be interpreted through

Kakutani's criterion:

The assumption µ ⊥ ν by Kakutani's criterion is equivalent to

∞∑n=1

d−1∑i=0

(√p

(n)i −√pi

)2

=∞

It follows that there exists α ∈ 0, 1, . . . , d− 1 such that

∞∑n=1

(√p

(n)α −√pα

)2

=∞

41

The non-singularity of T by Kakutani's criterion is equivalent to

∞∑n=1

d−1∑i=0

(√p

(n+1)i −

√p

(n)i

)2

<∞

and in particular ∣∣∣p(n+1)i − p(n)

i

∣∣∣ −−−−→n→∞

0

for every i ∈ 0, 1, . . . , d− 1.

The conservativity of T implies that∑∞N=1 ρ

(µ, µ TN

)= ∞ where

ρ(µ, µ TN

)is the Hellinger distance, as we showed above. We claim that

this implies that there exists a subsequence (nk)∞k=1 for which p

(nk)i −−−−→

k→∞pi for every i ∈ 0, 1, . . . , d− 1.Assume by contradiction that there exists ε > 0 and n0 ∈ N, such that for

every n ∈ N there exists i ∈ 0, 1, . . . , d− 1 with∣∣∣∣√p(n0+n)

i −√pi∣∣∣∣ ≥ ε.

Then for every N > n0,

d(µ, µ TN

)=∑∞n=1

∑d−1i=0

(√p

(n−N)i −

√p

(n)i

)2

≥∑Nn=n0

∑d−1i=0

(√pi −

√p

(n)i

)2

≥ (N − n0) · ε2

Thus, using the approximation we mentioned above,

ρ(µ, µ TN

)≤ C · exp

(−d(µ, µ TN

)2) ≤ C · exp(−Nε2

)contradiction to

∑∞N=1 ρ

(µ, µ TN

)=∞.

Under this properties of the sequence(p

(n)0 , p

(n)1 , . . . , p

(n)d−1

)∞n=1

and (p0, p1, . . . , d− 1),

our assertion can be reformulated into the following:

Proposition. Let µ be a half-stationary product measure µ =∏n∈Z µn on X =

0, 1, . . . , d− 1Z, where µn = (p0, p1 . . . , pd−1) for every n ≤ 0, 0 < pi < 1, and

µn =(p

(n)0 , p

(n)1 , . . . , p

(n)d−1

)for every n > 0. Assume that the following hold:

1. There exists α ∈ 0, 1, . . . , d− 1 such that∑∞n=1

(p

(n)α − pα

)2

=∞.

2. There exists a subsequence (nk)∞k=1 for which p

(nk)i −−−−→

k→∞pi for every

i ∈ 0, 1, . . . , d− 1.

3.∣∣∣p(n+1)i − p(n)

i

∣∣∣ −−−−→n→∞

0 for every i ∈ 0, 1, . . . , d− 1.

42

Then e (O, ψ) = R.

Denition. By assumption 2 in the proposition we can x a subsequence

(Nn)∞n=1 such that

∑∞n=1

(√p

(Nn)i −√pi

)2

<∞ for every i ∈ 0, 1, . . . , d− 1.

Let (Mn)∞n=1 be the sequence of all positive integers outside (Nn)

∞n=1. For a

later purpose we choose (Nn)∞n=1 such that Nn+1 − Nn > 1. This means that

Mn −Mn+1 ≤ 2 soMn+1 ∈ Mn + 1,Mn + 2

for every n, and by assumption 3∣∣∣p(Mn+1)i − p(Mn)

i

∣∣∣ −−−−→n→∞

0.

Dene µ =∏n∈Z µn by µNn = (p0, p1, . . . , pd−1) and µMn

= µMnfor every n.

We see that∑∞n=1

∑d−1i=0

(√µn (i)−

√µn (i)

)2

=∑∞n=1

∑d−1i=0

(√µNn (i)−

√µNn (i)

)2

=∑∞n=1

∑d−1i=0

(√p

(Nn)i −√pi

)2

<∞

so by Kakutani's criterion µ ∼ µ. Since orbit-equivalence is invariant of equiva-lence of measures, we assume WLOG that µ = µ; That is, p

(Nn)i = pi for every

n and i.

Proposition. For the α as in the proposition above, there exists a subsequence(nm)

∞m=1 such that the following properties hold:

1. p(Mnm )α −−−−→

m→∞pα

2.∑∞m=1

(√p

(Mnm )α −√pα

)2

=∞

3. Either p(Mnm )α ≥ pα for every m or p

(Mnm )α ≤ pα for every m.

Proof. Assume that the these properties do not hold for (Mn)∞n=1 itself. First

assume that qα is another partial limit of(p

(Mn)α

)∞n=1

with pα < qα. Recall

that∣∣∣p(Mn+1)α − p(Mn)

α

∣∣∣ −−−−→n→∞

0 as we mention in the above denition, and since

pα and qα are both partial limits of(p

(Mn)α

)∞n=1

it follows that the sequence(p

(Mn)α

)∞n=1

is dense in the interval (pα, qα). Thus we can construct a subse-

quence (nm)∞m=1 for which p

(Mnm )α ≥ pα for every m and p

(Mnm )α −−−−→

m→∞pα slow

enough so that∞∑m=1

(√p

(Mnm )α −√pα

)2

=∞

In the same way, if pα > qα we can construct (nm)∞m=1 with the same properties

but p(Mnm )α ≤ pα for every m.

43

Now assume that pα is the only partial limit. Then since∑∞n=1

(√p

(Mn)α −√pα

)2

=

∞ we can nd a subsequence (nm)∞m=1 for which p

(Mnm )α ≥ pα for every m and

or p(Mnm )α ≤ pα for every m, and

∞∑m=1

(√p

(Mnm )α −√pα

)2

=∞

Either way, there exists a subsequence with the desired properties.

Denition. Given the subsequence (nm)∞m=1 as in the above proposition, since∑d−1

i=0 pi = 1 =∑d−1i=0 p

(n)i for every n, there must be some β ∈ 0, 1, . . . , d− 1

such that either p(Mnm )β ≤ pβ for every m (if p

(Mnm )α ≥ pα for every m) or

p(Mnm )β ≥ pβ for every m (if p

(Mnm )α ≤ pα for every m).

We rewrite nm = m WLOG and dene the ip

(Rx)Nn = xMn, (Rx)Mn

= xNn ⇐⇒ xNn = β, xMn= α

for nitely many coordinates which we will specify later.

Claim. For the R we just dened we have

ψR = logR′

Proof. Assume that R changes the coordinates 1, . . . , N . We calculate bothfunctions separately.

As for ψR, since p(Nn)i = pi we see that for every x,

ψR (x) =∑Nn=1

∑d−1i=0

(1xn=i − 1(Rx)n=i

)log pi

p(n)i

=∑Nn=1

∑d−1i=0

(1xMn=i − 1(Rx)Mn=i

)log pi

p(Mn)i

For every n, if xNn = β, xMn= α then

d−1∑i=0

(1xMn=i − 1(Rx)Mn=i

)log

pi

p(Mn)i

= logpα

p(Mn)α

− logpβ

p(Mn)β

while otherwise it is zero. Thus,

ψR (x) =

N∑n=1

1xNn=α,xMn=β

(log

pα

p(Mn)α

− logpβ

p(Mn)β

)

As for logR′, we see that for every x,

logR′ (x) =∑Nn=1

∑d−1i=0

(1(Rx)Nn=i − 1xNn=i

)log pi

+∑Nn=1

∑d−1i=0

(1(Rx)Mn=i − 1xMn=i

)log p

(Mn)i

44

For every n, if xNn = β, xMn = α then

d−1∑i=0

(1(Rx)Nn=i − 1xNn=i

)log pi = log pα − log pβ

and

d−1∑i=0

(1(Rx)Mn=i − 1xMn=i

)log p

(Mn)i = log p

(Mn)β − log p(Mn)

α

while otherwise both terms are zero. Thus,

logR′ (x) =

N∑n=1

1xNn=α,xMn=β ·

(log

pα

p(Mn)α

− logpβ

p(Mn)β

)

which is the same formula we found for ψR (x).

Lemma. Consider the random variables,

Yn (x) = 1xNn=α,xMn=β ·

(log

pα

p(Mn)α

− logpβ

p(Mn)β

), SN1 (x) =

N∑n=1

Yn (x)

Then one of the following holds:

1. For every r > 0 there exists δ > 0 and N such that,

µ(SN1 > r

)≥ δ

2. For every r < 0 there exists δ > 0 and N such that,

µ(SN1 < r

)≥ δ

Proof. First we show that∣∣E [SN1 ]∣∣ −−−−→

N→∞∞. Note that

E [Yn] = pα · p(Mn)β ·

(log

pα

p(Mn)α

− logpβ

p(Mn)β

)≈ O

(log

pα

p(Mn)α

− logpβ

p(Mn)β

)

where this approximation means that the quotient of the two sides of the equa-tion tends to a positive constant as n → ∞. We consider the following twopossibilities:

pα ≥ p(Mn)α −−−−→

n→∞pα , pβ ≤ p(Mn)

β −−−−→n→∞

pβ (1)

pα ≤ p(Mn)α −−−−→

n→∞pα , pβ ≥ p(Mn)

β −−−−→n→∞

pβ (2)

45

In case (1) we have log pα

p(Mn)α

≥ 0 ≥ logpβ

p(Mn)β

and

logpα

p(Mn)α

− logpβ

p(Mn)β

≥ logpα

p(Mn)α

≈ O(p(Mn)α − pα

)In case (2) we have log pα

p(Mn)α

≤ 0 ≤ logpβ

p(Mn)β

and

logpα

p(Mn)α

− logpβ

p(Mn)β

≤ logpα

p(Mn)α

≈ O(p(Mn)α − pα

)and we also note that

p(Mn)α − pα ≈ O

(√p

(Mn)α −√pα

)where all the approximation are in the same sense as above.

Thus, by∑∞n=1

(√p

(Mn)α −√pα

)2

= ∞ we see that in case (1) we have

E[SN1]−−−−→N→∞

−∞ and in case (2) we have E[SN1]−−−−→N→∞

∞.

Using Paley-Zygmund inequality (See appendix A) for the r.v. SN1 we get

µ(SN1 > θE

[SN1])≥

(1− θ)2E[SN1]2

V(SN1)

+ (1− θ)2E[SN1]2

for every 0 < θ < 1. Calculation yields that

V (Yn) = pα · p(Mn)β

(1− pα · p(Mn)

β

)·(

log pα

p(Mn)α

− logpβ

p(Mn)β

)2

≈ C · p2α ·(p

(Mn)β

)2

·(

log pα

p(Mn)α

− logpβ

p(Mn)β

)2

= C ·E [Yn]2

where the approximation means that the quotient of the two sides of the equationtends to 1 as n→∞ for C > 0 the positive constant

C :=1− pα · pβpα · pβ

Now we get the following bound for large enough N ,

V(SN1)

+ (1− θ)2E[SN1]2 ≤ V

(SN1)

+ E[SN1]2 ≤ C ·∑N

n=1E [Yn]2

+ E[SN1]2

= (1 + C)∑Nn=1E [Yn]

2+∑n6=kE [Yn]E [Yk]

(∗) ≤ (1 + C)[∑N

n=1E [Yn]2

+∑n 6=kE [Yn]E [Yk]

]= (1 + C)E

[SN1]2

46

where the inequality (∗) holds since

log pα − log p(Mn)α − log pβ + log p

(Mn)β

is either non-negative (if p(Mn)α ≥ pα, p(Mn)

β ≤ pβ) or non-positive (in the oppo-site case). Thus E [Yn] ≥ 0 for every large n or E [Yn] ≤ 0 for every large n.Either way E [Yn]E [Yk] ≥ 0 for large n, k. By Kakutani's criterion, changingthe distribution of nitely many coordinates does not change the equivalenceclass of the measure as long as we do not give zero mass to any element of anycoordinate, so we can assume that either E [Yn] ≥ 0 for every n or E [Yn] ≤ 0for every n.In conclusion we get that for every 0 < θ < 1,

µ(SN1 > θE

[SN1])≥

(1− θ)2E[SN1]2

(1 + C)E[SN1]2 =

(1− θ)2

1 + C> 0

Now we are done: Fix θ = 1/2. If E[SN1]−−−−→N→∞

∞ then given any r > 0, for

large N we have 1/2 · E[SN1]> r and µ

(SN1 > r

)≥ 1

4(1+C) := δ. Thus the

case 1 in the lemma occurs. In the same way, if E[SN1]−−−−→N→∞

−∞ then using

Paley-Zygmund inequality for −SN1 yields that case 2 in the lemma occurs.

Remark. In the proof of the lemma we preferred to use Paley-Zygmund inequal-ity, a variation of Cauchy-Schwartz inequality, rather then the central limittheorem (CLT). See [DL, Prop. 1.5] for a use of CLT for a similar purpose.Though, it would be quite easy to use CLT to establish our lemma:For a sequenceW1,W2, . . . of uniformly bounded independent random variableswith zero mean, if

∑Nn=1 V (En) −−−−→

N→∞∞ then Lyapunov's condition holds and

we have CLT. Applying this to the r.v. Wn := Yn − E [Yn], as one can verifythat V

(SN1)−−−−→N→∞

∞ , we get that

µ(SN1 > E

[SN1])

= µ

(SN1 −E

[SN1]

V(SN1) > 0

)−−−−→N→∞

Φ (0) = 1/2

and the lemma follows in the same way as in our proof above.

Proof. (Main proposition). We show that if E[SN1]−−−−→N→∞

∞ then e (O, ψ) ⊃(0,∞). In the same way it will be clear that if E

[SN1]−−−−→N→∞

−∞ then

e (O, ψ) ⊃ (−∞, 0). Either way, since e (O, ψ) is an additive subgroup of Rit follows that e (O, ψ) = RUsing Choksi-Hawkins-Prasad lemma we mentioned above, we need to showthat for every 0 < ε < r there exists δ > 0 such that

µ

(⋃R

(C ∩R−1C ∩ |logR′ − r| < ε

))≥ δ · µ (C)

47

where R runs over the O-transformations (note that this is a Z-action denedby the odometer). We show that this holds for δ as in our lemma above.Let C be a set supported on nitely many coordinates. Let r > 0 and let0 < ε < r. Fix l large enough such that C is supported on 1, . . . , l − 1 and also|Yn| < ε for n > l − 1. By the lemma above there exists M > l and δ > 0 suchthat µ

(SMl > r

)≥ δ. Let D =

SMl > r

∩ C ⊂ C and since C and D are

supported on dierent coordinates we see that,

µ (D) = µ(SMl > r

)· µ (C) ≥ δ · µ (C)

For x ∈ D denote m (x) := min m ≤M | Sml (x) > r. Let R : D → C denedby

(Rx)Nn :=

xMn xNn = β, xMn = α, l ≤ Nn ≤ m (x)

xNn otherwise

(Rx)Mn:=

xNn xNn = β, xBn = α, l ≤Mn ≤ m (x)

xMnotherwise

for every n ∈ N.We claim that R is an O-transformation. Clearly R changes nitely manycoordinates. We have left to show that R is one-to-one. Assume Rx = Ry forx, y ∈ D with m (x) ≤ m (y). For Nn ∈ l, . . . ,m (x) we have xNn = (Rx)Mn

=(Ry)Mn

= yNn and forMn ∈ l, . . . ,m (x) we have xMn = (Rx)Nn = (Ry)Nn =yMn . Thus xn = yn for every 1 ≤ n ≤ m (x). In particular we see that

r > Sm(x)l (x) = S

m(x)l (y) and by denition of m (y) we see that m (x) ≥ m (y),

hence m (x) = m (y). Now by Rx = Ry we get that x = y.Finally, for every x ∈ D we see by denition of m (x) and by m (x) ≥ l that

r < Sm(x)l (x) = S

m(x)−1l (x) + Ym(x) (x) < r + ε

Since ψR (x) = Sm(x)l (x) we see that

|ψR (x)− r| = |logR′ (x)− r| < ε

and r ∈ e (O, ψ).

48

REFERENCES REFERENCES

References

[Aar] J. Aaronson. An Introduction to Innite Ergodic Theory, Amer. Math.Soc., Providence, R.I., (1997).

[Adl] Roy L. Adler. Symbolic dynamics and markov partitions. Bull. Am.Math. Soc.. New Ser. 35 (1998), no. 1, 156.

[ANS] J. Aaronson, H. Nakada, O. Sarig. Exchangeable measures for subshifts.Ann. Inst. H. Poincare Probab. Statist. 42 (2006), no. 6, 727-751.

[AW] R. Adler and B. Weiss. Entropy, a complete metric invariant for automor-phisms of the torus. Proc. Nat. Acad. of Science. 57 (1967). 1573-1576.

[Bo] V. I. Bogachev. Measure Theory. vol I. Springer (2007)

[Bow] R. Bowen. Markov partitions for Axiom A dieomorphisms. Amer. J.Math. 92 (1970), 725-747.

[CFW] A. Connes, J. Feldman, B. Weiss. An amenable equivalence relationis generated by a single transformation. Ergodic Theory & DynamicalSystems, vol. 1 (1981), 431-450.

[CHP] J. R. Choksi, J. M. Hawkins and V. S. Prasad. Abelian cocycles for non-singular ergodic transformations and the genericity of type III1 trans-formations. Monatsh. Math. 103 (1987) 187-205.

[DL] A. I. Danilenko and M. Lemanczyk. K-property for Maharam extensionsof nonsingular Bernoulli and Markov shifts. Ergodic Theory & Dynami-cal Systems. (to appear). Available at arxiv.org/abs/1611.05173.

[DS] A. I. Danilenko, C. E. Silva. Ergodic theory: Nonsingular transforma-tions. Mathematics of Complexity and Dynamical Systems. 329356.Springer (2011).

[FM] J. Feldman, C. C. Moore. Ergodic equivalence relations, cohomology andvon Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289324.

[GS] G. Greschonig, K. Schmidt. Ergodic decomposition of quasi-invariantprobability measures. Colloq. Math. 84/85 (2000), 495-514.

[Mah] D. Maharam. Incompressible transformations. Fund. Math. 56 (1964),3550.

[Ham] T. Hamachi. On a Bernoulli shift with non-identical factor measures.Ergodic Theory & Dynamical Systems, 1(3) (1981), 273-283.

[Haw] J. M. Hawkins. Amenable relations for endomorphisms. Trans. Amer.Math. Soc. 343 (1994), 169191.

49

http://arxiv.org/abs/1611.05173

REFERENCES REFERENCES

[HOO] T. Hamachi, Y. Oka, M. Osikawa. Flows Associated with Ergodic Non-Singular Transformation Groups. Publ. RIMS Kyoto Univ. (1975-1976),Volume 11, Issue 1, 31-50.

[Kak] S. Kakutani. On Equivalence of Innite Product Measures. Annals ofMathematics (1948), 2nd Series, 49, 214-224.

[KH] A. Katok, B. Hasselblatt. Introduction to the Modern Theory of Dy-namical Systems. Cambridge University Press (1995).

[KM] A. S. Kechris, B. D. Miller. Topics in orbit equivalence, volume 1852 ofLecture Notes in Mathematics. Springer-Verlag, Berlin (2004).

[Kos] Z. Koslo. On the K property for Maharam extensions of Bernoulli shiftsand a question of Krengel. Israel J. Math. 199 (2014), 485506.

[Kos1] Z. Koslo. On Manifolds Admitting Stable Type III1 Anosov Dieo-morphisms. Journal of Modern Dynamics. (to appear). Available atwww.math.huji.ac.il/∼zemkos/III_1_Anosov.pdf.

[KW] Y. Katznelson, B. Weiss. The classication of nonsingular actions, revis-ited. Ergodic Theory & Dynamical Systems, 11(2) (1991), 333348.

[Orn] D. S. Ornstein. On invariant measures. Bull. Amer. Math. Soc., Volume66, Number 4 (1960), 297-300.

[Sch] K. Schmidt. Cocycles on ergodic transformation groups. Macmillan Lec-tures in Mathematics, Vol. 1. (1977), Delhi.

[Shi] A.N. Shiryaev. Probability, 2nd ed., Graduate Texts in Mathematics Vol.95 (1996). Springer-Verlag, New York.

[ST] C. E. Silva, P. Thieullen. A skew product entropy for nonsingular trans-formations. J. London Math. Soc. 52 (1995), 497516.

[vonN] von Neumann, J. On rings of operators III. Annals of Mathematics,Second Series, 41 (1) (1940), 94161

[VW] S. Vaes and J. Wahl. Bernoulli actions of type III1 and L2-cohomology.Geometric and Functional Analysis 28 (2018), 518-562.

[We] B. Weiss. Measurable Dynamics. Contemp. Math., 26, (1984), 395-421.

50

http://www.math.huji.ac.il/~zemkos/III_1_Anosov.pdf

A APPENDIX: PALEY-ZYGMUND INEQUALITY

A Appendix: Paley-Zygmund Inequality

Let X be a random variable with nite variance σ2 := V (X) < ∞. Denoteµ := E [X]. Then for every 0 < θ < 1,

P (X > θµ) ≥ (1− θ)2µ2

σ2 + (1− θ)2µ2

Note that the right side is

E [X − θµ]2

E[(X − θµ)

2] =

µ2 − 2θµ2 + θ2µ2

E [X2]− 2θµ2 + θ2µ2

=µ2 − 2θµ2 + θ2µ2

σ2 + µ2 − 2θµ2 + θ2µ2=

(1− θ)2µ2

σ2 + (1− θ)2µ2

Hence, by rearranging terms the inequality becomes

E [X − θµ]2 ≤ E

[(X − θµ)

2]· P (X > θµ)

Now it follows from Cauchy-Schwartz inequality for L2:

E [X − θµ]2 ≤ E

[(X − θµ) · 1X>θµ

]2≤ E

[(X − θµ)

2]·E[1X>θµ

]= E

[(X − θµ)

2]· P (X > θµ)

51

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