Real-analytic realization of Uniform Circular Systems and

some applications

Shilpak Banerjee ∗ and Philipp Kunde †

April 2, 2020

Abstract

Recently Matthew Foreman and Benjamin Weiss showed in a series of papers that smooth er-godic diffeomorphisms of a compact manifold are unclassifiable up to measure-isomorphism. Inthis paper we show that the uniform circular systems used in the work of Foreman-Weiss admitreal-analytic realizations on the torus. As a consequence we obtain the same anti-classificationresult for real-analytic ergodic diffeomorphisms on the torus. In another application we showthe existence of an uncountable family of pairwise non-Kakutani equivalent real-analytic dif-feomorphisms on the torus.

Contents

1 Introduction 21.1 Statement of anti-classification results . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 82.1 Basics in Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Periodic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Real-analytic diffeomorphisms on the torus . . . . . . . . . . . . . . . . . . . . . . . 11

3 Symbolic systems 123.1 Symbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Odometer-based symbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Circular symbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Categories OB and CB and the functor F : OB → CB . . . . . . . . . . . . . . . . . 17

4 Abstract untwisted AbC method 184.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 The untwisted AbC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Special requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

∗Indraprastha Institute of Information Technology, Delhi, India. Portion of this paper were completed when S.B.was partially supported by NSF grant DMS-16-02409 at Pennsylvania State University†Pennsylvania State University, Department of Mathematics, State College, PA 16802, USA. P.K. acknowledges

financial support from a DFG Forschungsstipendium under Grant No. 405305501.

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Introduction 2

5 Approximating partition permutations by real-analytic diffeomorphisms 215.1 Block-slide type maps and their analytic approximations . . . . . . . . . . . . . . . . 215.2 Approximating partition permutation by diffeomorphisms . . . . . . . . . . . . . . . 22

6 Real-analytic AbC method 236.1 Real-analytic AbC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Fast enough in the real-analytic context . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Symbolic representation of AbC systems 267.1 Symbolic representation of periodic processes . . . . . . . . . . . . . . . . . . . . . . 277.2 Comparison of two periodic processes using a transect - I . . . . . . . . . . . . . . . 287.3 Comparison of two periodic processes using a transect - II . . . . . . . . . . . . . . . 297.4 Back to the regular sequence of periodic processes . . . . . . . . . . . . . . . . . . . 317.5 Symbolic representation of AbC systems . . . . . . . . . . . . . . . . . . . . . . . . . 31

8 Proof of Theorem E 378.1 Trees, groups and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 Specifications and timing assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 398.3 The continuous function F : T rees→MPT . . . . . . . . . . . . . . . . . . . . . . . 418.4 Numerical requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.5 F s = R F F is a continuous reduction . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Further applications 439.1 An uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomor-

phisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.2 Real-analytic non-Bernoulli diffeomorphisms with property K . . . . . . . . . . . . . 45

10 References 46

1 Introduction

In the foundational paper [Ne32] John von Neumann formulated the so-called isomorphism problemfor classifying the ergodic measure preserving transformations up to measure theoretic isomor-phisms, i.e. it asks to determine when two measure-preserving transformations are isomorphic. Ithas been a guiding light for directions of research within ergodic theory, and has been solved onlyfor some special classes of transformations, e.g.

• In 1942 von Neumann and Paul R. Halmos showed that the spectrum of the associatedKoopman operator is a complete isomorphism invariant for ergodic measure preserving trans-formations with pure point spectrum (see [HN42]).

• In 1970 Donald Ornstein showed that Bernoulli shifts are completely classified by their entropy(see [Or70]).

Many properties of transformations like mixing of various types or finite rank have been charac-terized and studied in connection with this problem but the general problem remained intractable.

Introduction 3

Starting in the late 1990’s, so-called anti-classification results have been established and demon-strated in a rigorous way that classification is not possible. This type of results requires a precisedefinition of what it means to obtain a classification result.

Informally, a classification is a method of determining isomorphism between transformations,perhaps by computing other invariants for which equivalence is easy to determine. Given a trans-formation (abstract or smooth or real-analytic) T defined on a measure space (or appropriatemanifold), one can ask whether or not it is possible to accurately describe the set U of all othertransformations (abstract or smooth or real-analytic) that are measure theoretically isomorphic toT . If a classification for such systems is available, then one can expect to proceed by computinginvariants and reducing a set containing T to smaller and smaller sets by requiring all elementsto have a uniform value of the invariant being considered. For example, one could start with a Tand look at the set containing all S with the same entropy as that of T and then in the next step,one could further restrict by requiring their Koopman operator to have the same set of eigenvaluesand so on and so forth. So if one wants to prove that certain transformations are not classifiable,one would need to show that no countable (possibly transfinite) protocol, whose basic input ismembership in open or closed sets exists that can be used to determine membership in a partic-ular equivalence class. So in other words, the equivalence relation should not be Borel. So, theBorel/non-Borel distinction is a natural and key notion in anti-classification results:

• In 1996 Ferenc Beleznay and Matthew Foreman showed that the class of measure distaltransformations used in early ergodic theoretic proofs of Szemeredi’s theorem is not a Borelset [BF96].

• In 2001 Greg Hjorth introduced the notion of turbulence and showed that there is no Borelway of attaching algebraic invariants to ergodic transformations that completely determineisomorphism [Hj01].

• Foreman and Benjamin Weiss improved Hjorth’s result by proving that the conjugacy actionof the measure preserving transformations is turbulent and hence no generic class can have acomplete set of algebraic invariants [FW04].

• Passing from a single transformation to pairs (S, T ) of measure preserving Hjorth [Hj01]showed that the equivalence relation defined by isomorphism is not a Borel set. Since hisproof uses nonergodic transformations in an essential way, this left open the question whathappens if one restricts to ergodic transformations. This was resolved by Foreman, Daniel J.Rudolph and Weiss who showed that the measure-isomorphism relation for ergodic measurepreserving transformations is also not a Borel set [FRW11].

Recently, in a series of papers ([FW19a], [FW19b] and [FW3pp]) Foreman and Weiss showed ananti-classification result for C∞ diffeomorphisms of compact manifolds by proving that the measure-isomorphism relation among pairs of volume preserving ergodic C∞-diffeomorphisms is not a Borelset with respect to the C∞-topology. Actually, it is a complete analytic set (see Definitions 1.2 and1.3).

The goal of this article is to upgrade the aforementioned anti-classification result to the real-analytic category on T2. At this juncture, we point out that the set of all volume preservingreal-analytic diffeomorphisms on a torus is not a metrizable space. In fact it is very difficult towork in this space and we prefer to do all construction on a certain subset Diff ω

ρ (T2, µ). Forsome pre-specified finite number ρ > 0 this subset consists of all volume preserving real-analytic

Introduction 4

diffeomorphisms homotopic to the identity whose lift to R2 allows a complexification extending to aband of width ρ in the imaginary direction (see Section 2.3). This set has the structure of a Polishspace and we can state the following theorem.

Theorem A. For every ρ > 0 the measure-isomorphism relation among pairs (S, T ) ∈ Diff ωρ (T2, µ)×

Diff ωρ (T2, µ) is a complete analytic set and hence not a Borel set.

Note that the space of all measure preserving real-analytic diffeomorphisms, Diff ω(T2, µ) :=∪ρ>0Diff ω

ρ (T2, µ) can be equipped with the direct limit topology, i.e. a set is U ⊂ Diff ω(T2, µ) isopen iff U ∩Diff ω

ρ (T2, µ) is open in Diff ωρ (T2, µ). See [Ly99, appendix 2] for a detailed discussion

of these spaces. However the space obtained in this way in not metrizable. We can see the followingresult as an immediate consequence of theorem A.

Theorem B. The measure-isomorphism relation among pairs (S, T ) ∈ Diff ω(T2, µ)×Diff ω(T2, µ)is not a Borel set, when Diff ω(T2, µ) is equipped with the direct limit topology.

The Foreman-Weiss anti-classification result and Theorem A are related to another major ques-tion in ergodic theory dating back to the pioneering paper [Ne32]: The smooth realization problemasks whether there are smooth versions to the objects and concepts of abstract ergodic theory andwhether every ergodic measure-preserving transformation has a smooth model. Here, by a smoothmodel one means a smooth diffeomorphism of a compact manifold preserving a measure equivalentto the volume element which is isomorphic to the measure-preserving transformation. The onlyknown restriction is due to A. G. Kushnirenko who proved that such a diffeomorphism must havefinite entropy. On the other hand, there is a lack on general results on the smooth realizationproblem.

One way to show that not all finite ergodic measure preserving transformation have a smoothmodel would be to show that their classification is easier than the general classification result. Butthe Foreman-Weiss anti-classification result shows that the variety of ergodic transformations thathave smooth models is rich enough so that the abstract isomorphism relation restricted to thesesmooth models is as complicated as it is in general. Theorem A shows that this even holds true forreal-analytic diffeomorphisms of T2.

One of the key steps in the recent work by Foreman and Weiss to adapt the methods from[FRW11] to the case of volume preserving C∞ diffeomorphisms is to show that a class of symbolicsystems, the so-called strongly uniform circular systems (see Section 3.3), can be realized as C∞

diffeomorphisms. To obtain these smooth realizations the so-called untwisted AbC method is used.This method was introduced by D.V. Anosov and A. Katok in their seminal paper [AK70] and it isalso widely known as the approximation by conjugation method or the Anosov-Katok method. Inthis paper we prove the real-analytic counterpart of the main result in [FW19a] that can be veryloosely summarized as the following theorem.

Theorem C. Let T be an ergodic transformation on a standard measure space. Then the followingare equivalent:

1. T is isomorphic to an real-analytic (untwisted) Anosov-Katok diffeomorphim (satisfying somerequirements).

2. T is isomorphic to a (strongly uniform) circular system (with fast growing parameters) .

Introduction 5

For an accurate version of the above theorem stated with all the required technicalities we referthe reader to Theorem G. Since this general realization of uniform circular systems as diffeomor-phisms reduces questions about diffeomorphisms to combinatorial questions for symbolic shifts, itbears a lot of flexibility to address questions in the realm of the smooth realization problem.

To exemplify the flexibility and strength of the real-analytic realization of circular systems weconstruct an uncountable family of real-analytic ergodic diffeomorphisms that are pairwise non-Kakutani equivalent. Recall that two ergodic transformations are said to be Kakutani equivalentif they are isomorphic to measurable cross-sections of the same ergodic flow. Then it immediatelyfollows from Abramovs entropy formula, that two Kakutani equivalent automorphisms must havethe same entropy type: zero entropy, finite entropy, or infinite entropy. It was a long-standing openproblem whether these three possibilities for entropy completely characterized Kakutani equivalenceclasses. Until the work of A. Katok [Ka75, Ka77] in the case of zero entropy, and J. Feldman [Fe76]in the general case, no other restrictions were known for achieving Kakutani equivalence. In [Fe76],Feldman showed that there are at least two non-Kakutani equivalent ergodic transformations ofentropy zero, and likewise for finite positive entropy and infinite entropy. Ornstein, Rudolph,and Weiss [ORW82] upgraded Feldman’s construction to obtain uncountably many non-Kakutaniequivalent ergodic MPT’s of each entropy type. Based on their construction, M. Benhenda [Be15]showed the existence of an uncountable family of pairwise non-Kakutani equivalent zero-entropyC∞ diffeomorphisms using the AbC method. As already indicated in [Ka03, Chapter 8], this resultalso follows from the smooth realization of uniform circular systems in general. We present thedetails in Section 9.1 and combine it with our real-analytic realization of uniform circular systemsto obtain the analogue in the real-analytic category.

Theorem D. For every ρ > 0 there exists an uncountable family of ergodic diffeomorphisms inDiff ω

ρ (T2, µ) which are pairwise not Kakutani-equivalent.

In Section 9.2 we use this result to construct real-analytic non-Bernoulli diffeomorphisms withproperty K on manifolds of dimension greater than 4. Note that a Bernoulli-automorphism (i. e. ameasure-preserving invertible transformation of a Lebesgue space isomorphic to a Bernoulli shift)has the K-property (i. e. a measure-preserving invertible transformation of a Lebesgue space obey-ing Kolmogorov’s zero-one law) automatically. Originally, Kolmogorov conjectured that the con-verse holds also true. The first counterexample in the measurable category was constructed byOrnstein [Or73]. First C∞ counterexamples were obtained by Katok in dimension greater than 4[Ka80]. Recently, A. Kanigowski, F. Rodriguez-Hertz, and K. Vinhage have found C∞ examples indimension 4 [KRV18]. On the other hand, smooth K-automorphisms in dimension 2 are Bernoulliby Pesin theory [Pe77].

The first real-analytic examples of non-Bernoulli diffeomorphisms with property K were ob-tained by Rudolph in [Ru88] as follows: Take a hyperbolic toral automorphism S on T2 given as(θ, η) : 0 ≤ θ, η < 1 and suppose Tt is the geodesic flow on the tangent space TM to a compacthyperbolic surface. Now define

S((θ, η), y) = (S(θ, η), Tsin(θ)(y)),

which is a real analytic K-diffeomorphisms and not Bernoulli.Actually, Theorem D enables us to show the existence of uncountably many real-analytic diffeo-

morphisms measure theoretically K, with the same entropy, and pairwise not isomorphic (not evenKakutani equivalent).

1.1 Statement of anti-classification results 6

1.1 Statement of anti-classification results

To state our anti-classification results precisely we will need some notions and concepts from De-scriptive Set Theory. The main tool is the idea of a reduction.

Definition 1.1. Let X and Y be Polish spaces and A ⊆ X, B ⊆ Y . A function f : X → Y reducesA to B if and only if for all x ∈ X: x ∈ A if and only if f(x) ∈ B.Such a function f is called a Borel (resp. continuous) reduction if f is a Borel (resp. continuous)function.

A being reducible to B can be interpreted as saying that B is at least as complicated as A. Wenote that if B is Borel and f is a Borel reduction, then A is also Borel. Taking the converse of thisstatement we get that if A is not Borel, then B is not Borel.

Definition 1.2. If S is a collection of sets and B ∈ S, then B is called complete for Borel (resp.continuous) reductions if and only if every A ∈ S is Borel (resp. continuously) reducible to B.

Following the interpretation above B is said to be at least as complicated as each set in S.

Definition 1.3. If X is a Polish space and B ⊆ X, then B is analytic if and only if it is thecontinuous image of a Borel subset of a Polish space. Equivalently, there is a Polish space Y and aBorel set C ⊆ X × Y such that B is the X-projection of C.

There are analytic sets that are not Borel. We use a canonical example of such a set: thecollection of ill-founded trees.

To introduce those, we consider the set N<N of finite sequences of natural numbers. A tree is aset T ⊆ N<N such that if τ = (τ1, . . . , τn) ∈ T and σ = (τ1, . . . , τs) with s ≤ n is an initial segmentof τ , then σ ∈ T . If σ is an initial segment of τ , then σ is a predecessor of τ and τ is a successorof σ. We define the level s of a tree T to be the collection of elements of T that have length s.

An infinite branch through T is a function f : N → N such that for all n ∈ N we have(f(0), . . . , f(n− 1)) ∈ T . If a tree has an infinite branch, it is called ill-founded. If it does not havean infinite branch, it is called well-founded.

In the following, let σn : n ∈ N be an enumeration of N<N with the property that every properpredecessor of σn is some σm for m < n. Under this enumeration subsets S ⊆ N<N can be identifiedwith characteristic functions χS : N → 0, 1. The collection of such χS can be viewed as the

members of on infinite product space 0, 1N<N

homeomorphic to the Cantor space. Here, eachfunction a : σm : m < n → 0, 1 determines a basic open set

〈a〉 = χ : χ σm : m < n = a ⊆ 0, 1N<N

and the collection of all such 〈a〉 forms a basis for the topology. The collection of trees is a closed

(hence compact) subset of 0, 1N<N

in this topology. Moreover, the collection of trees containingarbitrarily long finite sequences is a dense Gδ subset. In particular, this collection is a Polish space.We will denote the space of trees containing arbitrarily long finite sequences by T rees.

Since the topology on the space of trees was introduced via basic open sets giving us finiteamount of information about the trees in it, we can characterize continuous maps defined on T reesas follows.

Fact 1.4. Let Y be a topological space. Then a map f : T rees→ Y is continuous if and only if forall open sets O ⊆ Y and all T ∈ T rees with f(T ) ∈ O there is M ∈ N such that for all T ′ ∈ T reeswe have: if T ∩ σn : n ≤M = T ′ ∩ σn : n ≤M, then f (T ′) ∈ O.

1.2 Outline of the paper 7

As advertised the following classical fact (see e.g. [Ke95]) gives us an example of a completeanalytic set.

Fact 1.5. Let T rees be the space of trees.

1. The collection of ill-founded trees is a complete analytic subset of T rees.

2. The collection of trees that have at least two distinct infinite branches is a complete analyticsubset of T rees.

We also recall that the centralizer of an invertible measure preserving transformation T :(X,µ)→ (X,µ) is defined as C(T ) = S : X → X invertible m.p. transformation | S T = T S.Obviously, the powers T k belong to C(T ) for any k ∈ Z. We also stress that we consider the cen-tralizer in the group MPT of invertible measure preserving transformations and that C(T ) differsfrom the centralizer inside the group of diffeomorphisms.

Now we can present the precise statement of the main result of the paper that we will prove inSection 8.

Theorem E. For every ρ > 0 there is a continuous function F s : T rees→ Diff ωρ (T2, µ) such that

for T ∈ T rees, if T = F s(T ):

1. T has an infinite branch if and only if T ∼= T−1

2. T has two distinct infinite branches if and only if

C(T ) 6= Tn | n ∈ Z.

Then the classical Fact 1.5 implies the following anti-classification results.

Theorem F. For every ρ > 0 we have:

1.T ∈ Diff ω

ρ (T2, µ)∣∣ T ∼= T−1

is complete analytic.

2.T ∈ Diff ω

ρ (T2, µ)∣∣∣ C(T ) 6= Tn | n ∈ Z

is complete analytic.

Since the map i(T ) = (T, T−1) is a continuous mapping from Diff ωρ (T2, µ) to Diff ω

ρ (T2, µ) ×Diff ω

ρ (T2, µ) and reducesT∣∣ T ∼= T−1

to (S, T ) | S ∼= T, this result also yields that the measure-

isomorphism relation among pairs (S, T ) of ergodic Diff ωρ (T2, µ)-diffeomorphisms of T2 is a complete

analytic set and, hence, we obtain Theorem A as a consequence.

1.2 Outline of the paper

The theme of this paper is to make certain changes to the Foreman-Weiss’ series of paper in order toobtain the same anti-classification result for real-analytic diffeomorphisms of T2. In this connection,we also intend to give a survey of their impressive work.

In their proof of the anti-classification result for ergodic measure-preserving transformations in[FRW11] Foreman, Rudolph and Weiss construct a continuous function from the space T rees to theinvertible measure-preserving transformations assigning to each tree T a transformation T = F (T )of finite entropy such that T ∼= T−1 just in case T has an infinite branch (see Section 8.3). Thesetransformations have an odometer as a factor and, hence, are Odometer-based Systems in up-to-date

Preliminaries 8

terminology (see Section 3.2). Since it is a persistent open problem to find a smooth realization oftransformations with an odometer-factor, Foreman and Weiss circumvent that obstacle by showingthat the collection of Odometer-based Systems has the same global structure with respect to joiningsas another collection of transformations, the so-called Circular Systems that are extensions ofparticular circle rotations (see Section 3.3 for a precise description). For this purpose, they showin [FW19b] that there is a functor F between these classes that takes synchronous joinings tosynchronous joinings, anti-synchronous joinings to anti-synchronous joinings as well as synchronousand anti-synchronous isomorphisms to synchronous and anti-synchronous isomorphisms (see Section3.4 for an explanation of the terms synchronous and anti-synchronous).

The definition of these circular systems is inspired by a symbolic representation for circle rota-tions by certain Liouville rotation numbers found in [FW19a]. Then Foreman and Weiss showedthat these circular systems can be realized as volume preserving ergodic C∞-diffeomorphisms on atorus or a disk or an annulus (under some assumptions on the circular coefficients).

To obtain these smooth realizations the so-called untwisted AbC method is used. In their seminalpaper [AK70] D.V. Anosov and A. Katok introduced this method for constructing examples of C∞-diffeomorphisms. The construction can be carried out on any smooth compact connected manifoldthat admits an effective circle action R = Rtt∈S1 . It involves inductively constructing measurepreserving diffeomorphisms Tn = Hn Rαn H−1

n where the diffeomorphism Hn is obtained asthe compositions Hn = Hn−1 hn and the rational number αn is chosen close enough to αn−1 toguarantee convergence of the sequence Tn to a diffeomorphism T . Usually we want T to satisfysome dynamical property like weak-mixing, minimality, unique ergodicity, etc. and this is achievedby constructing hn at the n-th step in a way so that Tn satisfies some finite version of the targetedproperty.

Such constructions have resulted in several interesting results in the C∞-world. Beyond smooth-ness, the next natural question is the setting of real-analytic realizations. However, there are fewersuccess stories for real analytic diffeomorphisms due to certain well documented difficulties (seee.g. [FK04, section 7.1] as well as [BK19, Section 6.3]). In recent years new papers have ap-peared which demonstrate that such constructions are possible with enough flexibility on the torus([Sa03], [Ba17], [Ku17], [BK19]). A modified version of the construction also shows potential forreal-analytic constructions beyond the torus ([FK14]).

In the case of tori Td, d ≥ 2, the concept of block-slide type of maps introduced in [Ba17]and their sufficiently precise approximation by volume preserving real-analytic diffeomorphismsallows to find real-analytic counterparts of several Anosov-Katok constructions (see [BK19]). Thisapproach is the important mechanism in the constructions of this paper as well (we emphasize thatall constructions in this article are done on the torus and that real-analytic AbC constructions onarbitrary real-analytic manifolds continue to remain an intractable problem). Hereby, we get ourreal-analytic counterpart of the main result in [FW19a] in Theorem G that we already advertisedin a vague manner as Theorem C. Also, unlike Foreman-Weiss, we use T2 as our manifold and alsoas our abstract measure space. This is done to reduce notational complexity.

2 Preliminaries

Here we introduce the basic concepts and establish notations that we will use for the rest of thisarticle. We note that section 2.1 and 2.2 are standard theory presented from [FW19a, sections 3.1and 5] and hence we skip all proofs. Section 2.3 is presented with complete proofs since the theory

2.1 Basics in Ergodic Theory 9

is somewhat rare. However one can find similar or identical exposition in [Sa03],[FS05],[Ba17] and[BK19].

2.1 Basics in Ergodic Theory

We give an concise survey of some concepts regarding measure spaces and measure preservingtransformations. Our objective here is not to be comprehensive but rather to introduce some wellknown concepts in the context of our article and establish certain notations.

Let X be a set, B a Boolean algebra of measurable sets on X and m a measure. Then if thetriplet (X,B,m) is a separable non-atomic probability space, we call it a standard measure space.Let (X,B,m) and (X′,B′,m′) be two measure space. Then a map f defined on a set of full mmeasure of X onto a set of full m′ measure of X′ is called a measure theoretic isomorphism if f isone-one and both f and f−1 are measurable.

An ordered countable partition or simply a partition of (X,B,m) will refer to a sequence P :=Pn∞n=1 such that the following conditions are satisfied:

1. Pn ∈ B for all n ∈ N.

2. Pn ∩ Pm = ∅ if n 6= m.

3. m(∪∞n=1Pn) = 1.

Each Pn is called an atom of the partition P. Often we deal with partitions where all but finitelymany atoms have measure zero and we refer to such a partition as a finite partition. We say that asequence of partitions Pn∞n=1 is a generating sequence if the smallest σ-algebra containing ∪∞n=1Pnis B.

Next we introduce the notion of a distance between two partitions. Let P = Pn∞n=1 andQ = Qn∞n=1 be two partitions, we define

Dm(P,Q) :=

∞∑i=1

m(Pi4Qi) (2.1)

Lemma 2.2. Fix a sequence εn∞n=1 such that∑∞n=1 εn < ∞. Let (X,B,m) and (X′,B′,m′) be

two standard measure spaces and Tn∞n=1 and T ′n∞n=1 be measure preserving transformations ofX and X′ converging weakly 1 to T and T ′ respectively. Suppose Pn∞n=1 is a decreasing sequenceof partitions and Kn∞n=1 is s sequence of measure preserving transformations such that

1. Kn : X→ X′ is an isomorphism between Tn and T ′n.

2. Pn∞n=1 and Kn(Pn)∞n=1 are generating sequence of partitions for X and X′.

3. Dm(Kn+1(Pn),Kn(Pn)) < εn.

Then the sequence Kn converges in the weak topology to a measure theoretic isomorphism betweenT and T ′.

1 The set of all measure preserving transformation on X has a natural topology where a basic open set is givenby the sets N(T,P, ε) := S : S is a measure preserving transformation and

∑A∈P m(TA4SA) < ε for a measure

preserving transformation T , a finite measurable partition P and some ε > 0. This is called the weak topology.

2.2 Periodic processes 10

2.2 Periodic processes

Here we recall some relevant facts from the notion of periodic processes. One can refer to [Ka03]for a detailed account. We continue with notations from the previous subsection.

Let P be a partition of (X,B,m) where all the atoms of P have the same measure. A Periodicprocess is a pair (τ,P) where τ is a permutation of P such that each cycle has equal length 2. Werefer to these cycles as towers and their length is called height of the tower. We also choose anatom from each tower arbitrarily and call it the base of the tower. In particular if P1, . . . ,Ps arethe towers (of height q) of this periodic process with B1, . . . , Bs as their respective bases, then anytower Pi can be explicitly written as Bi, τ(Bi), . . . , τ

q−1(Bi). We refer to τk(Bi) as the k-th levelof the tower Pi. τ q−1(Bi) is the top level. Next we describe how to compare two periodic processes.

Definition 2.3 (ε-approximation). Let (τ,P) and (σ,Q) be two periodic processes of the measurespace (X,B,m). We say that (σ,Q) ε-approximates (τ,P) if there exists disjoint collections of Qatoms SA : A ∈ P, SA ⊂ Q. and a set D ⊂ X of measure less than ε such that the following aresatisfied:

1. For every A ∈ P, we have ∪B : B ∈ SA \D ⊂ A.

2. If A ∈ P is not the top level of a tower and B ∈ SA, we have σ(B) \D ⊆ τ(A)

3. For each tower of σ, the measure of the intersection of X \D with each level of this tower arethe same.

With the above definition in mind, we have the following result regarding the convergence ofperiodic processes.

Lemma 2.4. Let εnn∈N be a summable sequence of positive numbers. Let (τn,Pn)n∈N be asequence of periodic processes on (X,B,m) such that (τn+1,Pn+1) εn-approximates (τn,Pn) and thesequence Pnn∈N is generating.

Then there exists a unique transformation T : X→ X satisfying:

limn→∞

m(⋃

A∈Pn

(τnA4TA)) = 0 (2.5)

We call (τn,Pn) a convergent sequence of periodic processes.

Lemma 2.6. Let (τn,Pn)n∈N be a sequence of periodic processes converging to T and Tnn∈Nbe a sequence of measure preserving transformations satisfying for each n,∑

A∈Pn

µ(TnA4τnA) < εn (2.7)

Then Tnn∈N converges weakly to T .

Lemma 2.8. Let εnn∈N be a summable sequence of positive numbers. Let (X1,B1,m1) and(X2,B2,m2) be two standard measure spaces. Let Tnn∈N and Snn∈N be two sequences of measurepreserving transformations of X1 and X2 converging to T and S respectively in the weak topology.Suppose Pn is a decreasing sequence of partitions and φnn∈N be a sequence of measure preservingtransformations such that

2 This is not a necessary requirement for the definition, but is good enough for us.

2.3 Real-analytic diffeomorphisms on the torus 11

1. φn is a measure theoretic isomorphism between Tn and Sn.

2. The sequence Pn and φn(Pn) generate B1 and B2.

3. Dm2(φn+1(Pn), φn(Pn)) < εn

Then the sequence φn converges in the weak topology to an isomorphism between T and S.

2.3 Real-analytic diffeomorphisms on the torus

We will denote the two dimensional torus by

T2 := R2/Z2 (2.9)

We use µ to denote the standard Lebesgue measure on T2.We give a description of the space of measure preserving real-analytic diffeomorphisms on T2

that are homotopic to the identity.Any real-analytic diffeomorphism on T2 homotopic to the identity admits a lift to a map from

R2 to R2 and has the following form

F (x1, x2) = (x1 + f1(x1, x2), x2 + f2(x1, x2)) (2.10)

where fi : R2 → R are Z2-periodic real-analytic functions. Any real-analytic Z2 periodic functionon R2 can be extended as a holomorphic (complex analytic) function from some open complexneighborhood 3 of R2 in C2. For a fixed ρ > 0, we define the neighborhood

Ωρ := (z1, z2) ∈ C2 : |Im(z1)| < ρ and |Im(z2)| < ρ (2.11)

and for a function f defined on this set, put

‖f‖ρ := sup(z1,z2)∈Ωρ

|f((z1, z2))| (2.12)

We define Cωρ (T2) to be the space of all Z2-periodic real-analytic functions on R2 that extends toa holomorphic function on Ωρ and ‖f‖ρ <∞.

We define, Diff ωρ (T2, µ) to be the set of all measure preserving real-analytic diffeomorphisms of

T2 homotopic to the identity, whose lift F to R2 satisfies fi ∈ Cωρ (T2) and we also require that the

lift F (x) = (x1 + f1(x), x2 + f2(x)) of its inverse to R2 to satisfy fi ∈ Cωρ (T2).The metric in Diff ω

ρ (T2, µ) is defined by

dρ(f, g) = maxdρ(f, g), dρ(f−1, g−1) where dρ(f, g) = max

i=1,2 infn∈Z‖fi − gi + n‖ρ

Next, with some abuse of notation, we define the following two spaces

Cω∞(T2) := ∩∞n=1 Cωn (T2) (2.13)

Diff ω∞(T2, µ) := ∩∞n=1 Diff ω

n(T2, µ) (2.14)

We now list some properties of the above spaces that are going to be useful to us.

3We identify R2 inside C2 via the natural embedding (x1, x2) 7→ (x1 + i0, x2 + i0).

Symbolic systems 12

• Note that the functions in 2.13 can be extended to C2 as entire functions.

• Diff ω∞(T2, µ) is closed under composition. To see this assume that f, g ∈ Diff ω

∞(T2, µ) and letF,G be their lifts to R2. Then note that F G is the lift of the composition f g (with π : R2 →T2 as the natural projection, πF G = f πG = f gπ). Now for the complexification of Fand G note that the composition F G(z) = (z1+g1(z)+f1(G(z)), z2+g2(z)+f2(G(z))). Sincegi ∈ Cω∞(T2), we have for any ρ, supz∈Ωρ |Im(G(z))| ≤ maxi(supz∈Ωρ |Im(zi) + Im(gi(z))|) ≤maxi(supz∈Ωρ |Im(zi)|+supz∈Ωρ |Im(gi(z))|) ≤ ρ+maxi(supz∈Ωρ |gi(z)|) < ρ+const < ρ′ <∞for some ρ′. So, supz∈Ωρ |gi(z) + fi(G(z))| ≤ supz∈Ωρ |gi(z)| + supz∈Ωρ |fi(G(z))| < ∞ since

gi ∈ Cω∞(T2), G(z) ∈ Ωρ′ and fi ∈ Cωρ′(T2). An identical treatment gives the result for theinverse.

• If fn∞n=1 ⊂ Diff ωρ (T2, µ) is Cauchy in the dρ metric, then fn converges to some f ∈

Diff ωρ (T2, µ). Indeed, uniform convergence guarantees analyticity and consideration of the

inverses ensures that the result is a diffeomorphism. So this space is a Polish space 4.

Similarly we define the space of all real-analytic functions on T2 and all measure preservingreal-analytic diffeomorphisms of T2 as follows:

Cω(T2) := ∪∞n=1 Cω1n

(T2) (2.15)

Diff ω(T2, µ) := ∪∞n=1 Diff ω1n

(T2, µ) (2.16)

The space of all measure preserving real-analytic diffeomorphisms of T2 is given the correspond-ing inductive limit topology and it is not a metrizable space. This space is usually very difficult towork with and we will not use this for the rest of this article.

This completes the description of the analytic topology necessary for our construction. Alsothroughout this paper, the word “diffeomorphism” will refer to a real-analytic diffeomorphismunless stated otherwise. Also, the word “analytic topology” will refer to the topology of Diff ω

ρ (T2, µ)described above. See [Sa03] for a more extensive treatment of these spaces.

3 Symbolic systems

In this section we introduce the notion of a symbolic system in a way that is most convenientin representing AbC transformations. In particular we write about the uniform circular systemsintroduced by Foreman and Weiss. We skip proofs and recall the results only for one can refer to[FW19a, sections 3.3 and 4] for the details. We also note that we try to stick to the notations usedin [FW19a] but we do make some changes.

3.1 Symbolic systems

Let Σ be a finite or countable alphabet endowed with the discrete topology. By ΣZ we denote thespace of bi-infinite sequences of alphabets from Σ endowed with the product topology. The producttopology makes this a totally disconnected separable space that is compact if Σ is finite.

4 A Polish space is a separable completely metrizable topological space

3.1 Symbolic systems 13

In order to get a better description of the of the product topology on this set we define for anyu = 〈u0, u1, . . . , un−1〉 ∈ Σ<∞,

Ck(u) := f ∈ ΣZ : f |[k,k+n) = u

Such sets are known as cylinder sets and they generate the product topology of ΣZ. Next we definethe (left) shift map

sh : ΣZ → ΣZ defined by sh(f)(n) = f(n+ 1) (3.1)

If µ is a shift invariant Borel measure then the system (ΣZ,B, µ, sh) is called a symbolic system. Theclosure of the support of µ is a shift invariant measure preserving system and we call it a symbolicshift or a sub shift.

We recall that we can construct symbolic shifts from an arbitrary measure preserving transfor-mation (X,B, µ, T ). To see this, fix a partition P := Ai : i ∈ I for some countable or finite Iand an alphabet Σ := ai : i ∈ I. We define φ : X → ΣZ by φ(x)(n) = ai ⇐⇒ Tnx ∈ Ai. Theφ∗µ is an invariant measure and (ΣZ, C, φ∗µ, sh) is a factor of (X,B, µ, T ) with factor map φ. If Pis generating then φ is an isomorphism.

The above description of symbolic shift and coding is the most straight forward way, but toexplicitly understand the resulting symbolic shift conjugate with an AbC transformation we needa step by step inductive procedure for describing symbolic shifts which in certain way emulatesthe AbC process. So we give an intrinsic definition of the symbolic shifts using the notion ofconstruction sequences.

Definition 3.2. A sequence of collection of words (Wn)n∈N, where N = 0, 1, 2, . . . , satisfying thefollowing properties is called a construction sequence:

1. for every n ∈ N all words in Wn have the same length hn,

2. each w ∈Wn occurs at least once as a subword of each w′ ∈Wn+1,

3. there is a summable sequence (εn)n∈N of positive numbers such that for every n ∈ N, everyword w ∈ Wn+1 can be uniquely parsed into segments u0w1u1w1 . . . wlul+1 such that eachwi ∈ Wn, each ui (called spacer or boundary) is a word in Σ of finite length and for thisparsing ∑l+1

i=0|ui|hn+1

< εn+1.

Next we define the following subset of ΣZ:

K := x ∈ ΣZ : if x = uwv for some u, v ∈ ΣZ, w ∈ Σ<∞ then w

is a contiguous subword of some w′ ∈Wn for some n (3.3)

We note that K is a closed shift invariant subset of ΣZ.Before we talk about measures on K, we need some technical definitions. We say that a con-

struction sequence Wn is uniform if there exists a sequence of functions dn : Wn → (0, 1)∞n=1 suchthat for some summable sequence of positive numbers εn∞n=1, we have for any choice of w ∈ Wn

and w′ ∈Wn+1,∣∣∣ hnhn+1

(#i : w = wi where w′ = u0w0u1w1 . . . ulwlul+1)− dn(w)∣∣∣ < 1

hnεn+1 (3.4)

3.2 Odometer-based symbolic systems 14

We say that K is a uniform symbolic system if it is build out of a uniform construction sequence.If it so happens that the number #i : w = wi where w′ = u0w0u1w1 . . . ulwlul+1 depends

only on n but not on w or w′ then we refer to the construction sequence and K as strongly uni-form. Note that strong uniformity implies uniformity with dn(w) = #i : w = wi where w′ =u0w0u1w1 . . . ulwlul+1(hn/hn+1).

We say that a collection of finite words W is uniquely readable iff whenever u, v, w ∈ W anduv = pws then either p or s is the empty word.

Let K be a uniform symbolic system built out of a construction sequence Wn where each Wn isuniquely readable. We define

S := x ∈ K : ∃ natural number sequences am∞m=1, bm∞m=1 satisfying x|[am,bm) ∈Wm (3.5)

and note that S is a dense shift invariant Gδ set.

Lemma 3.6. Let K be a uniform symbolic system built out of the construction sequence Wn∞n=1

in a finite alphabet Σ. Then the following holds:

1. K is the smallest shift invariant closed subset of ΣZ such that

K ∩ C0(w) 6= ∅ ∀n ∈ N w ∈Wn (3.7)

2. There exists a unique non-atomic shift invariant measure ν concentrated on S and this measureis ergodic.

3. For any s ∈ S ⊂ K and w ∈Wn, the density of k : w occurs in s starting at k exists and isequal to ν(C0(w)). Moreover ν(C0(w)) = dn(w)/hn.

3.2 Odometer-based symbolic systems

Let (kn)n∈N be a sequence of natural numbers kn ≥ 2 and

O =∏n∈N

(Z/knZ)

be the (kn)n∈N-adic integers. Then O has a compact abelian group structure and hence carries aHaar measure λ. We define a transformation T : O → O to be addition by 1 in the (kn)n∈N-adicintegers (i.e. the map that adds one in Z/k0Z and carries right). Then T is a λ-preserving invertibletransformation called odometer transformation which is ergodic and has discrete spectrum.

We now define the collection of symbolic systems that have odometer systems as their timingmechanism to parse typical elements of the system.

Definition 3.8. Let (Wn)n∈N be a uniquely readable construction sequence with W0 = Σ and

Wn+1 ⊆ (Wn)kn for every n ∈ N. The associated symbolic shift will be called an odometer-based

system.

Thus, odometer-based systems are those built from construction sequences (Wn)n∈N such thatthe words in Wn+1 are concatenations of a fixed number kn of words in Wn. Hence, the words in Wnhave length hn, where

hn =

n−1∏i=0

ki

3.3 Circular symbolic systems 15

if n > 0, and h0 = 1. Moreover, the spacers in part 3 of Definition 3.2 are all the empty words (i.e.an odometer-based transformation can be built by a cut-and-stack construction using no spacers).

Remark. According to anannouncement in [FW19b], any finite entropy system that has an odometerfactor can be represented as an odometer-based system

3.3 Circular symbolic systems

We now describe a special class of symbolic shifts called circular symbolic systems. These systemswere introduced by Foreman and Weiss in [FW19a, section 4] and are specifically designed to serveas symbolic representations of untwisted AbC systems. We recall the results from their work butskip most proofs.

Consider natural numbers k, l, p, q with p and q mutually prime. We note that for every i =0, 1, . . . , q − 1, there exists a 0 ≤ ji < q such that the following holds:

pji = i mod q (3.9)

we often use the notation

ji = (p)−1i mod q (3.10)

for brevity. Note that q − ji = jq−i.In order to describe uniform circular systems, we start by defining an operator on alphabets.

Let Σ be any alphabet and b, e be two letter not in Σ. Suppose w0, . . . , wk−1 be a sequence ofwords constructed out of the alphabet Σ ∪ b, e. We define the operator:

C(w0, w1, . . . , wk−1) :=

q−1∏i=0

k−1∏j=0

(bjq−iwl−1j eji) (3.11)

Note that

• If |wi| = q for i = 0, . . . , k − 1, then |C(w0, . . . , wk−1)| = klq2.

• For every e ∈ C(w0, . . . , wk−1), there exists a b to the left of it.

• If for some m > n, there exists a b at the n th position followed by a e at the m th position andadditionally we know that neither occurs inside a wi then there must exist a wi in betweenthe m th and the n th position.

• The proportion of the word w written in equation 3.11 that belongs to the boundary is 1/l.Moreover the proportion of the word that is within q letters of the boundary is 3/l.

We also introduce some notions around this map. Suppose w = C(w0, w1, . . . , wk−1). So wconsists of blocks with l − 1 copies of wi along with some b s and e s at the ends which are notinside wi. We refer to the portion of w in wi s to be the interior of w and the b s and e s not inthe wi s to be the boundary of w. In a block of the form wl−1

i , the first and the last occurrencesof wi is called the boundary portion of the block wl−1

j and the other occurrences are called interioroccurrences.

3.3 Circular symbolic systems 16

Lemma 3.12. Let Σ be a finite or countable alphabet and u0, . . . , uk−1, v0, . . . , vk−1, w0, . . . , wk−1

are words of length q < l/2 constructed from the alphabet Σ ∪ b, e. Put u = C(u0, . . . , uk−1), v =C(v0, . . . , vk−1) and w = C(w0, . . . , wk−1). If for some words p and s constructed from Σ ∪ b, ewe have

uv = pws

Then either p = empty word, u = w, v = s or s = empty word, u = p, v = w.

Next we inductively define four sequences of natural numbers kn∞n=1, ln∞n=1, pn∞n=1 Definek0, l0 and qn∞n=1 as follows:

pn+1 = pnqnknln + 1 qn+1 = knlnq2n (3.13)

We note that the above relations makes pn and qn relatively prime. So we can define for 0 ≤ i < qna natural number 0 ≤ ji < qn such that

ji = (pn)−1i mod qn (3.14)

Now we are ready to build a construction sequence for our symbolic shift. We choose a nonemptyfinite or countable alphabet Σ and choose two letters b and e not in Σ. We start or induction byputting W0 = Σ. Now we assume that the induction has been carried out till the n th step.

At the n+ 1 th step we we choose a set Pn+1 ⊂ (Wn)kn and put

Wn+1 := C(w0, . . . , wkn−1) : (w0, . . . , wkn−1) ∈ Pn+1 (3.15)

It follows from lemma 3.12 that Wn+1 is uniquely readable.Next we introduce the concept of strong unique readability. We can view Wn as a collection

of Λn letters and elements of Pn+1 can be viewed as words constructed out of Λn. If Pn+1 isuniquely readable in the alphabet Λn, we say that the construction sequence satisfies the strongunique readability assumption.

A construction sequence which satisfies 3.15, uses parameters satisfying 3.13, and satisfies thestrong unique readability assumption is called a circular construction sequence.

Lemma 3.16. If the ln∞n=1 parameters of a circular construction sequence satisfies

∞∑n=1

1/ln <∞ (3.17)

and for each n there exists a number fn such that each word w ∈ Wn occurs exactly fn times ineach word in Pn+1 then the circular construction sequence is strongly uniform.

Definition 3.18. A symbolic shift K constructed from a circular construction system is called acircular system. A symbolic shift K constructed from a (strongly) uniform circular constructionsystem is called a (strongly) uniform circular system.

Lemma 3.19. Let K be a circular system. Then

1. A shift invariant measure ν concentrates on S ⊂ K iff ν concentrates on the collection ofs ∈ K such that i : s(i) 6∈ b, e is unbounded in both Z+ and Z− direction.

3.4 Categories OB and CB and the functor F : OB → CB 17

2. If K is a uniform circular system and ν is a shift invariant measure on K, then ν(S) = 1. Inparticular, there is a unique non-atomic shift-invariant measure on K by Lemma 3.7.

We end this section after defining a canonical factor of a circular system measure theoreticallyisomorphic to a rotation of the circle. Let kn∞n=1 and ln∞n=1 be two parameter sequencessatisfying 3.17. Let K be any circular system constructed out of these parameters. We also constructa second circular system. Let Σ0 = ∗ and we define a construction sequence

W0 := Σ0, and if Wn := wn then Wn+1 = C(wn, . . . , wn). (3.20)

We denote the resulting circular system by K.Now we claim that K is a factor of K. We see that one can construct an explicit factor map as

follows

π : K→ K defined by π(x) :=

x(i) if x(i) ∈ b, e∗ otherwise

(3.21)

We end this section with the following observations:

• π : K→ K is Lipschitz.

• π sh± = sh± π.

• π is a factor map of K to K and from K−1 to K−1.

In [FW19b, Section 4.3] a specific isomorphism \ : K → rev(K) is introduced. It is called thenatural map and will serve as a benchmark for understanding of maps from K to rev(K) (see e.g.Definition 3.22).

3.4 Categories OB and CB and the functor F : OB → CBFor a fixed circular coefficient sequence (kn, ln)n∈N we consider two categories OB and CB whoseobjects are odometer-based and circular systems respectively. The morphisms in these categoriesare (synchronous and anti-synchronous) graph joinings.

Definition 3.22. If K is an odometer based system, we denote its odometer base by Kπ and letπ : K→ Kπ be the canonical factor map. Similarly, if Kc is a circular system, we let (Kc)π be therotation factor K and let π : Kc → K be the canonical factor map.

1. Let K and L be odometer based systems with the same coefficient sequence and let ρ bea joining between K and L±1. Then ρ is called synchronous if ρ joins K and L and theprojection of ρ to a joining on Kπ × Lπ is the graph joining determined by the identity map.The joining ρ is called anti-synchronous if ρ joins K and L−1 and the projection of ρ to ajoining on Kπ × (L−1)π is the graph joining determined by the map x 7→ −x.

2. Let Kc and Lc be circular systems with the same coefficient sequence and let ρ be a joiningbetween Kc and (Lc)±1. Then ρ is called synchronous if ρ joins Kc and Lc and the projectionof ρ to a joining on K × L is the graph joining determined by the identity map. The joiningρ is called anti-synchronous if ρ joins Kc and (Lc)−1 and the projection of ρ to a joining onK × L−1 is the graph joining determined by the map rev(·) \.

Abstract untwisted AbC method 18

In [FW19b] Foreman and Weiss define a functor taking odometer-based systems to circular sys-tem that preserves the factor and conjugacy structure. To review the definition of the functor wefix a circular coefficient sequence (kn, ln)n∈N. Let Σ be an alphabet and (Wn)n∈N be a construc-tion sequence for an odometer-based system with coefficients (kn)n∈N. Then we define a circularconstruction sequence (Wn)n∈N and bijections cn : Wn →Wn by induction:

• Let W0 = Σ and c0 be the identity map.

• Suppose that Wn, Wn and cn have already been defined. Then we define

Wn+1 = Cn (cn (w0) , cn (w1) , . . . , cn (wkn−1)) : w0w1 . . . wkn−1 ∈ Wn+1

and the map cn+1 by setting

cn+1 (w0w1 . . . wkn−1) = Cn (cn (w0) , cn (w1) , . . . , cn (wkn−1)) .

In particular, the prewords are

Pn+1 = cn (w0) cn (w1) . . . cn (wkn−1) : w0w1 . . . wkn−1 ∈ Wn+1 .

Definition 3.23. Suppose that K is built from a construction sequence (Wn)n∈N and Kc has thecircular construction sequence (Wn)n∈N as constructed above. Then we define a map F from theset of odometer-based systems (viewed as subshifts) to circular systems (viewed as subshifts) by

F (K) = Kc.

Remark. The map F is a bijection between odometer-based symbolic systems with coefficients(kn)n∈N and circular symbolic systems with coefficients (kn, ln)n∈N that preserves uniformity. Sincethe construction sequences for our odometer-based systems will be uniquely readable, the corre-sponding circular construction sequences will automatically satisfy the strong readability assump-tion.

In [FW19b] it is shown that F gives an isomorphism between the categories OB and CB. Westate the following fact which is part of the main result in [FW19b].

Fact 3.24. For a fixed circular coefficient sequence (kn, ln)n∈N the categories OB and CB areisomorphic by the functor F that takes synchronous isomorphisms to synchronous isomorphismsand anti-synchronous isomorphisms to anti-synchronous isomorphisms.

4 Abstract untwisted AbC method

We recall the AbC method in a way that is most convenient to us. The exposition is similar to theone in [FW19a, section 6.1] and [Ka03, Part I section 8].

Let R be the usual action of the circle T1 on T2 obtained by translation of the first coordinate.More precisely,

Rt : T2 → T2 defined by Rt(x1, x2) = (x1 + t, x2) (4.1)

4.1 Notations 19

4.1 Notations

We introduce some partitions of the circle T1. For any natural number q we define the partition ofthe unit circle into half open intervals of length 1/q as follows,

Iq :=[ iq,i+ 1

q

)⊂ T : i = 0, 1, . . . , q − 1

(4.2)

Next, given natural numbers s and q, we define a partition of the torus T2 into rectangles of length1/q and height 1/s as follows,

ξsq := Iq ⊗ Is :=[ iq,i+ 1

q

)×[js,j + 1

s

)⊂ T : i = 0, 1, . . . , q − 1; j = 0, 1, . . . , s− 1

(4.3)

In the AbC construction we deal with specific sequences of natural numbers qn and sn and we willoften use the following notation for convenience,

ξn := ξqn,sn (4.4)

and the respective atoms of the above partition will be denoted by

Rni,j :=[ iqn,i+ 1

qn

)×[ jsn,j + 1

sn

)(4.5)

We note that with α = p/q, p and q mutually prime, the atoms of ξsq is permuted by the action ofRα. This action results in a permutation consisting of s cycles, each of length q.

4.2 The untwisted AbC method

Now we give a description of the abstract ‘untwisted’ AbC method. This is an inductive process andwe assume that the construction has been carried out till the n th stage. So we have the followinginformation available to us,

1. Sequences of natural numbers kmn−1m=1, smnm=1, lmn−1

m=1, pmnm=1, qmnm=1 and a se-quence of rational numbers αmnm=1 satisfying the following relations

pm+1 = pmqmkmlm + 1 qm = kmlmq2m αm = pm/qm skmm ≥ sm+1 (4.6)

2. Measure preserving transformations hmnm=1 of the torus T2. We assume that hm is a permu-

tation of ξsmkm−1qm−1, hm commutes with R1/qm−1 and hm leaves the rectangle ∪sm−1−1

j=0 Rm−1i,j

invariant. 5

3. Measure preserving transformations Hmnm=1 and Tmnm=1 of the torus T2 satisfying thefollowing conditions,

Hm := h0 h1 . . . hm Tm := Hm Rαm H−1m (4.7)

5 The last condition is why we call this version of Anosov-Katok untwisted.

4.3 Special requirements 20

Now we describe how the construction is carried out in the n+ 1 the stage of the AbC method.We choose a natural number sn+1 followed by the natural number kn so that the following growthcondition is satisfied

sknn ≥ sn+1 (4.8)

Next we choose a measure preserving transformation hn+1 of T2 which is a permutation of ξknqn,sn+1

(and hence a permutation of ξn+1). Since we are doing the untwisted version of the AbC method,we additionally ensure that the transformation hn+1 leaves the rectangle [0, 1/qn) × T invariant.We finally choose ln to be a large enough natural number so that the conjugacies Tn and Tn+1 areclose enough and the sequence Tn converges in the weak topology. We also note that pn+1, qn+1

and αn+1 are automatically determined by the formulae pn+1 = pnqnknln + 1, qn+1 = knlnq2n and

αn+1 = pn+1/qn+1. This completes the description of our version of the abstract AbC method.Now we define the following sequence of partitions,

ζn = Hn(ξn) (4.9)

and note that since hm′ is a permutation of ξm for all m′ ≤ m, we can conclude that ζn is onlya permutation of ξn and hence is a generating sequence of finite partitions if ξn is a generatingsequence of partitions.

4.3 Special requirements

Till now we described the abstract untwisted AbC method in its full generality. In our case wewould need it to satisfy the following additional requirements:

• Requirement 1: The sequence sn tends to ∞

• Requirement 2: For each Rn0,j ∈ ξn and each s < sn+1, we havet < kn : hn+1

([ t

knqn,t+ 1

knqn

)×[ s

sn+1,s+ 1

sn+1

))⊂ Rn0,j

=knsn

(4.10)

Note that this assumption allows us to define a map s 7→ (j0, . . . , jkn−1)s from 0, 1, . . . , sn+1−1 to 0, 1, . . . , sn − 1kn so that for any fixed s < sn+1,

hn+1

([ t

knqn,t+ 1

knqn

)×[ s

sn+1,s+ 1

sn+1

))⊂ Rn0,jt

• Requirement 3: We assume that the map s 7→ (j0, . . . , jkn−1)s is one to one.

Note that the above requirements are more than enough to guarantee ergodicity for the limittransformation T . We end this section by stating an obvious lemma which serves as a converse tothe above.

Lemma 4.11. Let w0, . . . , wsn+1−1 ⊂ 0, 1, . . . , sn−1kn be words such that each i with 0 ≤ i ≤ snoccurs kn/sn times in each wj. Then there is an invertible measure preserving hn+1 commutingwith Rαn and inducing a permutation of ξknqn,sn+1 such that if jt is the t th letter of ws then

hn+1

([ t

knqn,t+ 1

knqn

)×[ s

sn+1,s+ 1

sn+1

))⊂ Rn0,jt

Approximating partition permutations by real-analytic diffeomorphisms 21

5 Approximating partition permutations by real-analytic dif-feomorphisms

The purpose of this section is to show that any permutation of a partition of T2 by a rectangular gridcan be approximated sufficiently well by real-analytic diffeommorphisms. This is the real-analyticversion of the content of [FW19a, section 6.2]. We note that their smooth construction can be doneon the torus, annulus or the disk. Unfortunately the lack of bump functions in the real-analyticcategory makes life harder and our real-analytic constructions are only valid for the torus. For adisk, even very basic questions like the existence of real-analytic ergodic diffeomorphisms remainopen. One can refer to [FK04, section 7.1] and [BK19, section 6.3] for a comprehensive analysis ofknown difficulties for a real-analytic AbC method on arbitrary real-analytic manifolds.

5.1 Block-slide type maps and their analytic approximations

We recall that a step function on the unit interval is a finite linear combination of indicator functionson intervals. We define the following two types of piecewise continuous maps on T2,

h1 : T2 → T2 defined by h1(x1, x2) := (x1, x2 + s1(x1) mod 1) (5.1)

h2 : T2 → T2 defined by h2(x1, x2) := (x1 + s2(x2) mod 1, x2) (5.2)

where s1 and s2 are step functions on the unit interval. The first map has the same effect aspartitioning T2 into smaller rectangles using vertical lines and sliding those rectangles verticallyaccording to s1. On the other hand the second map has the same effect as partitioning T2 intosmaller rectangles using horizontal lines and sliding those rectangles horizontally according to s2.We refer to any finite composition of maps of the above kind as a block-slide type of map on T2.This is somewhat similar to playing a game of nine without the vacant square on T2.

The purpose of the section is to demonstrate that a block-slide type of map can be approximatedextremely well by measure preserving real analytic diffeomorphisms. This can be achieved becausestep function and be approximated well by real analytic functions. We have the following lemmawhere we achieve this approximation and a little more to guarantee periodicity with a pre-specifiedperiod.

Lemma 5.3. Let k and N be two positive integer and α = (α0, . . . , αk−1) ∈ [0, 1)k. Consider astep function of the form

sα,N : [0, 1)→ R defined by sα,N (x) =

kN−1∑i=0

αiχ[ ikN ,

i+1kN )(x)

Here αi := αj where j := i mod k. Then, given any ε > 0 and δ > 0, there exists a periodicreal-analytic function sα,N : R→ R satisfying the following properties:

1. Entirety: The complexification of sα,N extends holomorphically to C.

2. Proximity criterion: sα,N is L1 close to sα,N . In fact we can say more,

supx∈[0,1)\F

|sα,N (x)− sα,N (x)| < ε, (5.4)

where F = ∪kN−1i=0 Ii ⊂ [0, 1) is a union of intervals centred around i

kN , i = 1, . . . , kN − 1 and

I0 = [0, δ2kN ] ∪ [1− δ

2kN , 1) and λ(Ii) = δkN ∀ i.

5.2 Approximating partition permutation by diffeomorphisms 22

3. Periodicity: sα,N is 1/N periodic. More precisely, the complexification will satisfy,

sα,N (z + n/N) = sα,N (z) ∀ z ∈ C and n ∈ Z (5.5)

4. Bounded derivative: The derivative is small outside a set of small measure,

supx∈[0,1)\F

|s′α,N (x)| < ε (5.6)

Proof. See [Ba17, Lemma 4.7] and [Ku17, Lemma 3.6].

Note that the condition 5.5 in particular implies

supz:Im(z)<ρ

sβ,N (z) <∞ ∀ ρ > 0.

Indeed, for any ρ > 0, put Ω′ρ = z = x + iy : x ∈ [0, 1], |y| < ρ and note that entirety of sβ,Ncombined with compactness of Ω′ρ implies supz∈Ω′ρ

|sβ,N (z)| < C for some constant C. Periodicity of

sα,N in the real variable and the observation Ωρ = ∪n∈Z(Ω′ρ + n

)implies that supz∈Ωρ |sβ,N (z)| <

C. We have essentially concluded that sβ,N ∈ Cω∞(T1).Now we show that block-slide type of maps on T2 can be approximated well by entirely extend-

able real-analytic diffeomorphisms.

Proposition 5.7. Let h : T2 → T2 be a block-slide type of map which commutes with φ1/q for somenatural number q. Then for any ε > 0 and δ > 0 there exists a measure preserving real-analyticdiffeomorphim h ∈ Diff ω

∞(T2, µ) such that the following conditions are satisfied:

1. Proximity property: There exists a set E ⊂ T2 such that µ(E) < δ and supx∈T2\E ‖h(x) −h(x)‖ < ε.

2. Commuting property: h φ1/q = φ1/q h

In this case we say the the diffeomorphism h is (ε, δ)-close to the block-slide type map h.

Proof. See [BK19, Proposition 2.22]

5.2 Approximating partition permutation by diffeomorphisms

Now we describe how partition permutations of T2 can be approximated well enough by real-analytic diffeomorphisms. At this juncture we point out that for the purpose of the AbC methodwe would like the approximating diffeomorphism constructed here to commute with R1/q for somegiven natural number q. In the smooth category this is achieved by carrying out the constructionon a fundamental domain of R1/q in such a way that this map is identity near the boundary of thisfundamental domain and then we glue together q translated copies of this diffeomorphism and theresulting diffeomorphism commutes with R1/q by construction.

In the real-analytic category we do not know how to reproduce the result in a similar fashionand hence we do all construction on the whole of T2 rather than a fundamental domain of T2. Theproblem if we do the construction this way is that we have to keep track that of commutativity allalong our construction.

Real-analytic AbC method 23

We briefly describe how this construction can be achieved using notations from section 4. Wefix natural numbers k, s, q and Π a permutation of ξskq which commutes with R1/q.

In [BK19, section 5.1] we proved the subsequent statement on the approximation of arbitrarypermutations.

Proposition 5.8 ([BK19], Theorem E). Let k, q, l ∈ N and Π be any permutation of kql elements.We can naturally consider Π to be a permutation of the partition Skq,l of the torus T2. Assumethat Π commutes with φ1/q. Then Π is a block-slide type of map.

We have the approximate real-analytic version of the above theorem as follows:

Theorem 5.9. Let Π be any permutation of k × s rectangles which partitions [0, 1/q) × T1. Inparticular we can extend this π to a permutation of ξskq which commutes with φ1/q (see proposition

5.8). Then for any ε > 0, there exists a diffeomorphism h ∈ Diff ω∞(T2, µ) such that for a set

L ⊂ T2, the following conditions are satisfied:

1. µ(L) > 1− ε.

2. For any x ∈ L ∩R, h(x) ∈ Π(R) for any R ∈ ξsq .

3. φ1/q h = h φ1/q.

We say that h ε-approximates Π.

Proof. Follows using proposition 5.8 followed by proposition 5.7.

We note that the above theorem is the real-analytic counterpart of [FW19a, Theorem 35] and it isvalid for the torus only. This is essentially the main difference of our paper with the Foreman-Weisswork. We also remark that one can further improve upon the above result and obtain the real-analytic analogue of [AK70, Theorem 1.2] and a reduced version of [AK70, Theorem 1.1] withoutthe boundary condition. Of course one already has real-analytic version of these theorems (Moser’stheorem is true in the real-analytic category) but the complexification of such diffeomorphisms arenot known to be entire in any sense and hence these are not compatible with the AbC method. Sothe above theorem and the aforementioned generalizations are the best we can hope for at the time.

6 Real-analytic AbC method

Our objective in this section is to produce examples of real-analytic diffeomorphisms using the AbCmethod. We will also show that these real-analytic AbC diffeomorphisms we construct here areisomorphic to the abstract AbC transformations constructed earlier.

6.1 Real-analytic AbC method

Theorem 6.1 (Real-analytic untwisted AbC diffeomorphisms). Fix a number ρ > 0. SupposeT : T2 → T2 is a measure preserving transformation built by the abstract AbC method usingparameter sequences kn∞n=1 and ln∞n=1.

Then if ln∞n=1 is a sequence of numbers which grows fast enough (see 6.9), there exists adiffeomorphism T (a) ∈ Diff ω

ρ (T2, µ) which is measure theoretically isomorphic to T .

6.1 Real-analytic AbC method 24

Proof. Fix εn∞n=1 such that∑∞n=1 εn < ∞. Let hn∞n=1, Hn = h1 . . . hn∞n=1 and Tn =

Hn Rαn H−1n ∞n=1 be a sequence of transformations constructed using parameters kn∞n=1 and

ln∞n=1 via the abstract AbC method.

Using theorem 5.9 we construct diffeomorphisms h(a)n ∈ Diff ω

∞(T2, µ) such that h(a)n εn-approximates

hn. We put

H(a)n := h

(a)1 . . . h(a)

n T (a)n := H(a)

n Rαn (H(a)n )−1 (6.2)

Now we make the following observation regarding proximity exploiting the commutation relation:

dρ(T

(a)n+1, T

(a)n

)= dρ

(H(a)n Rαn [h

(a)n+1 R1/(knlnq

2n) (h

(a)n+1)−1] (H(a)

n )−1, H(a)n Rαn (H(a)

n )−1)

Recall that ln is chosen last in the induction step. So, if one choses ln to be a large enough natural

number then from the continuity of dρ with respect to composition we obtain dρ(T(a)n+1, T

(a)n ) <

εn. Since we are dealing with real-analytic functions which are often more delicate than smooth

functions, we make some more observations to justify this claim. Note that since (H(a)n+1)−1 ∈

Diff ω∞(T2, µ), we can choose some ρ′ > ρ such that (H

(a)n+1)−1(Ωρ) ⊂ Ωρ′ . For any x ∈ Ωρ we put

y = (H(a)n+1)−1(x). So y, R1/(knlnq

2n)(y) ∈ Ωρ′ . Also note that H

(a)n+1 Rαn ∈ Diff ω

∞(T2, µ) and anyfunction in Diff ω

∞(T2, µ) is uniformly continuous on Ωρ′ .6 This implies

dρ(T(a)n+1, T

(a)n ) ≤ dρ′(H(a)

n+1 Rαn R1/(knlnq2n), H

(a)n+1 Rαn)

≤ supy∈Ωρ′

‖H(a)n+1 Rαn(R1/(knlnq

2n)(y)), H

(a)n+1 Rαn(y)‖

< εn/4 (6.3)

for ln sufficiently large. A similar consideration with the inverse gives us the result for dρ. Hence

the sequence T(a)n is Cauchy and converges to some T (a) ∈ Diff ω

ρ (T2, µ).

Next we need to prove that T (a) is in fact measure theoretically isomorphic to T . Our plan isto use lemma 2.2 for the proof.

In the language of the lemma, we put (X,B,m) = (X′,B′,m′) = (T2,B, µ). Next we define Kn.We put

Kn : T2 → T2 defined by Kn := H(a)n H−1

n (6.4)

From the definition it follows that Kn is a isomorphism between Tn and T(a)n .

We define the two sequences of partitions Pn := ζn = Hn(ξn) and P ′n := Kn(ζn) = H(a)n (ξn)

and observe that using lemma 4.11 we can conclude that Pn∞n=1 is generating. We have to worka little to show that P ′n is generating.

6 Indeed, if F ∈ Diff ω∞(T2, µ), then since F is Z2 periodic, we have for any ρ′′ > 0, u, v ∈ z = (z1, z2) ∈

C2 : Re(zi) ∈ [0, 1], Im(zi) ≤ ρ′′ and ε > 0, there exists a δ > 0 independent of u, v such that ‖u − v‖ < δ ⇒‖F (u) − F (v)‖ < ε (this follows from the compactness of the domain). In other words F is uniformly continuouson this restricted domain. For arbitrary u, v ∈ Ωρ′′ , is δ is small, we can find integers n1, n2,m1,m2 such that

u − (n1, n2), v − (m1,m2) ∈ z = (z1, z2) ∈ C2 : Re(zi) ∈ [0, 1], Im(zi) ≤ ρ′′. And since F is Z2 periodic,‖F (u)− F (v)‖ = ‖F (u− (n1, n2))− F (v − (m1,m2))‖.

6.1 Real-analytic AbC method 25

We define the set Ln to be the set of measure more than 1 − εn corresponding to hn as perTheorem 5.9. Consider the following sequence of sets:

Gn := Ln ∩∞⋂

m=n+1

(h(a)n+1 . . . h(a)

m )−1(Lm)

Note that Gn is an increasing sequence and the Borel-Cantelli lemma guarantees that µ(Gn) 1.We pick a measurable D ⊂ T2 and δ > 0. There exists some n0 such that µ(Gm) > 1 − δ

2 forall m > n0.

We put D′ = (H(a)n0 )−1(D) and since ξn∞n=1 is a generating sequence, there exists an m > n0

and a collection C′m ⊂ ξm such that

µ((⋃

C∈C′m

C)4D′)< δ/2

⇒µ((⋃

C∈C′m

H(a)n0

(C))4D)< δ/2

On the other hand, note that for any m > n0

µ( ⋃R∈ξm

h(a)n0+1 . . . h(a)

m (R) 4 hn0+1 . . . hm(R))< δ/2

⇒ µ( ⋃R∈ξm

(H(a)n0 h(a)

n0+1 . . . h(a)m (R) 4 H(a)

n0 hn0+1 . . . hm(R)

)< δ/2

⇒ µ( ⋃R∈ξm

(H(a)m (R) 4 H(a)

n0 hn0+1 . . . hm(R)

)< δ/2

⇒ µ( ⋃R∈ξm

(H(a)m (R) 4 H(a)

n0(Π(R))

)< δ/2

Where Π := hn0+1 . . . hm is a permutation of ξm. So,

µ( ⋃R∈Π−1(C′m)

H(a)m (R) 4 D

)≤ µ

( ⋃R∈Π−1(C′m)

H(a)m (R) 4 H(a)

n0(Π(R))

)+ µ

( ⋃R∈Π−1(C′m)

H(a)n0

(Π(R)) 4 D)

≤ µ( ⋃R∈ξm

H(a)m (R) 4 H(a)

n0(Π(R))

)+ µ

( ⋃C∈C′m

H(a)n0

(C) 4 D)

≤ δ/2 + δ/2

= δ

This shows that P ′n is a generating sequence of partitions.Our next objective is to show that Dµ(Kn+1(Pn),Kn(Pn)) < εn. So we do the following

computations:

Kn(Pn) = Kn(ζn) = H(a)n H−1

n (Hn(ξn)) = H(a)n (ξn) = H(a)

n hn+1 h−1n+1(ξn)

6.2 Fast enough in the real-analytic context 26

On the other hand,

Kn+1(Pn) = Kn+1(ζn) = H(a)n+1 H

−1n+1(Hn(ξn)) = H(a)

n h(a)n+1 h

−1n+1(ξn)

Put Qn := h−1n+1(ξn) and note that by construction h

(a)n+1(Qn) approximates hn+1(Qn) and we are

done.

6.2 Fast enough in the real-analytic context

We investigate what it means to be fast enough in theorem 6.1. We fix a sequence εn∞n=1 andassume that for any n ∈ N:

εn/4 >

∞∑m=n

εm (6.5)

For each choice of sequences knnm=1, lnn−1m=1 and snn+1

m=1 of natural numbers, we can havefinitely many permutations of ξ

sn+1

knqnand hence finitely many choices of hn+1. For each such choice,

there exists a natural number ln := ln(hn+1, knnm=1, lnn−1m=1, sn

n+1m=1, ρ) such that for any l ≥ ln,

we can choose h(a)n+1 such that

dρ(T(a)n , T

(a)n+1) < εn/4 (6.6)

Finally to get an uniform estimate, we put

l∗n = l∗n(knnm=1, lnn−1m=1, sn

n+1m=1, ρ) := max

hn+1

ln(hn+1, knnm=1, lnn−1m=1, sn

n+1m=1, ρ) (6.7)

Now we note that for our construction there is a relation sn = skn−1

n−1 . We can define bn =

bn(knn−1m=1) such that sn < bn. Thus we can have our sequence l∗n depend only on kn s (af-

ter speeding up a little if needed).In conclusion we obtain a sequence of natural numbers

l∗n = l∗n(knnm=1, lnn−1m=1, ρ) (6.8)

and we say a sequence of natural numbers ln∞n=1 grows fast enough if

ln ≥ l∗n ∀ n. (6.9)

7 Symbolic representation of AbC systems

The purpose of this section is establish the main result where a symbolic representation of untwistedreal-analytic AbC diffeomorphisms is obtained. This section is almost identical to [FW19a, section7] though our presentation is much succinct.

We begin this section with the intrinsic description of obtaining a symbolic representation ofthe factor of a periodic process using construction sequences.

7.1 Symbolic representation of periodic processes 27

7.1 Symbolic representation of periodic processes

Here we exhibit how a periodic process can be viewed as a symbolic system. Everything in thissection is done on a standard measure space (X,B,m).

Let εnn∈N be a summable sequence of positive numbers. Let (τn,Pn) be a sequence ofperiodic processes converging to some transformation T . We assume that the height of all thetowers of (τn,Pn) is qn and (τn+1,Pn+1) εn-approximates (τn,Pn) with error set Dn. In additionwe can assume (after removing some steps in the beginning if needed) that µ(∪Dn) < 1/2 and alsoassume q0 = 1.

We put Gn = X \ ∪m≥nDn. Then note that Gnn∈N is an increasing sequence of sets withµ(Gn) 1. Also, when restricted to Gn, Pmm≥n is a decreasing sequence of partitions. For eachtower of (τn,Pn), the measure of the intersection of Gn with each level of this tower are all thesame.

Let Tis0−1i=0 be the towers of (τ0,P0. Since q0 = 1, each Ti contains a single set and we define

Q0 to be the collection of all these sets.We inductively define two sequence of sets Bnn∈N and Enn∈N. Let B0 = E0 = ∅. Next we

assume that we have defined Bmnm=0 and Emnm=0.At the n + 1 th stage of the induction process we have to define Bn+1 and En+1. Since τn+1 εn-approximates (τn,Pn), we note that each tower of (τn+1,Pn+1) when restricted to Gn+1 contains

1. Contiguous levels: Contiguous sequences of levels of length qn contained in towers of (τn,Pn)

2. Interspersed levels: Levels of the towers of (τn+1,Pn+1) intersected with Gn+1 not in theabove contiguous sequences.

Next we divide each of the maximal contiguous portions of the interspersed levels into two contiguousportions arbitrarily. We define En+1 to be the union of En and the subcollection of levels that comesfirst and Bn+1 to be the union of Bn and the subcollection that comes second. Also we note the topand the bottom contiguous subcollection of interspersed levels are joined together and is consideredto be a single contiguous subcollection for this process.This completes the construction of the two sequences and we note that Gn = Q0 ∪Bn ∪ En.

Our next goal is to find a symbolic representation of a factor of the limiting transformationT arising from the partition Q0 ∪ ∪n∈NBn,∪n∈NEn. We do this in two ways: First we describethe standard procedure using T and then we describe the procedure that uses the aforementionedsequences and construction sequence description of symbolic systems.

Let Σ be an alphabet of size s0. Let b and e be two letters not contained in Σ. The lettersin Σ := ais0−1

i=0 are considered to be indexes for the elements of Q0 := Ais0−1i=0 . Our symbolic

systems is built on the alphabet Σ ∪ b, e. We define a factor map

K : X→ (Σ ∪ b, e)Z defined by K(x)(i) :=

aj if T i(x) ∈ Ajb if T i(x) ∈ ∪n∈NBne if T i(x) ∈ ∪n∈NEn

(7.1)

We give an alternate description of this symbolic system using construction sequences. Weinductively define Wn which are collections of words of length qn and surjections Kn : T ∩Gn : Tis a tower of (τn,Pn) and Gn ∩ T 6= ∅ → Wn.

First we put W0 := Σ and we assume that we have carried out the construction upto the n thstage. At the n + 1 th stage we define the collection of words Wn+1 and Kn+1. Let T be a towerof (τn+1,Pn+1). Then with T ∩Gn+1 we associate a word w of length qn+1 satisfying:

7.2 Comparison of two periodic processes using a transect - I 28

1. w(j) = v if the j-th level of T is a subset of the k-th level of of a tower S of (τn,Pn) and thek-th letter of Kn(S) is v.

2. w(j) = b if the j-th level of T is a subset of Bn+1 \Bn.

3. w(j) = e if the j-th level of T is a subset of En+1 \ En.

We define K to be the collection of all x ∈ (Σ ∪ b, e)Z such that every contiguous subword of xis a contiguous subword of some w ∈ Wn for some n. Then K is a closed shift invariant set thatconstitutes the support of K∗µ and it is the required symbolic representation of the factor of Tdescribed explicitly earlier (see lemma 3.6).

We note that in the case of the untwisted AbC transformations we consider satisfying require-ments 1, 2 and 3 (see section 4.3), Q will generate the transformation and the resulting symbolicrepresentation will be isomorphic to T .

7.2 Comparison of two periodic processes using a transect - I

Instead of directly comparing two subsequent periodic processes in the AbC method, we start ourstudy slowly with the study of two periodic processes on the circle. This study will shed light onhow a tower in the periodic process at the n-th stage of the AbC method compares with a towerat the n + 1-th stage. In fact we wish to study how the levels of a periodic process on the circletraverses the levels of another periodic process which it ε-approximates. The parameters for theprocesses are chosen similar to the AbC method.

Let p, q be two natural numbers and α = p/q. Iq := Iqi := [i/q, (i+ 1)/q)q−1i=0 is the standard

partition of [0, 1) into q equal half open intervals. Recall that the atoms of Iq have two naturalorderings:

1. The geometric ordering is the natural ordering of the intervals i.e. Iq0 < Iq1 < . . . < Iqq−1.

2. The dynamical ordering is the ordering that comes from iteration by Rα i.e. Iq0 < Rα(Iq0 ) <. . . < (Rα)q−1(Iq0 ).

Recall that with ji = (p)−1i mod q, the i-th interval in the geometric ordering is the ji th intervalin the dynamical ordering. Indeed if (Rα)ji(Iq0 ) = Iqi′ , then pji ≡ i′.

Let α = p/q and α′ = p′/q′ where p, p′, q and q′ are natural numbers and q′ = klq2. We comparethe two periodic processes (Rα, Iq) and (Rα

′, Iq′).

So if J is a subinterval of Iqt , and if J is not the last subinterval of Iqt then Rα′(J) is a subinterval

of the jith subinterval in the dynamical ordering i.e. Rα(Iqt ). If J is the last subinterval, then Rα′(J)

is geometrically the first subinterval of the ji+1-th subinterval in the dynamical ordering.With the above observation in mind, we make detailed analysis of how the interval J := [0, 1/q′)

traverses the levels of the tower of (Rα, Iq) under the action of Rα′. Note that we have to study q′

iterates of Rα′

to get a complete picture. So we divide the set 0, 1, . . . , q′ = klq2 into contiguousportions of size q.

On the first portion i.e. for n = 0, 1, . . . , klq − 1, (Rα′)n(J) ⊂ (Rα)n(Iq0 ) and (Rα

′)klq(J) is

geometrically the first subinterval of Iq1 .More generally when we study the iterations of J for n = mklq, . . . , (m+ 1)klq, we see that the

iterations exhibit the following three types of behavior:

7.3 Comparison of two periodic processes using a transect - II 29

1. The beginning interval [mklq, . . . ,mklq + q − jm): This is of length q − jm. Note that(Rα

′)mkql(J) is geometrically the first subinterval of Iqm. Then with n in the beginning

interval, (Rα′)n(J) traverses the interval in places jm, jm + 1, . . . , q − 1 in the dynamical

ordering of (Rα, Iq).

2. The middle interval [mkql+q− jm,mkql+q− jm+(kl−1)q): This is of length klq−q. Withn in the middle interval, (Rα

′)n(J) traverses the intervals following the dynamical ordering

of (Rα, Iq) starting from Iq0 .

3. The end interval [mkql + q − jm + (kl − 1)q, (m + 1)kql): This has length jm. With n inthe end interval, (Rα

′)n(J) traverses the interval in places 0, 1, . . . , jm − 1 in the dynamical

ordering of (Rα, Iq). (Rα′)mklq−1(J) is geometrically the last subinterval of Iqm.

7.3 Comparison of two periodic processes using a transect - II

The previous section did shed some light on how two periodic processes compare on the circlewhen one ε approximates the other but we note that in the previous section we did not pay muchattention to the fact that the AbC method uses the partition whose projection is Ikq and not Iq.We do the study again, but keeping this finer partition in mind.

Let p, q, p′, q′, α and α′ be as before. We divide the subinterval Ikq into k ordered sets describedas follows:

wj := Ikqj+tk : t = 0, . . . , q − 1 (7.2)

So each wj is the orbit of [j/(kq), (j + 1)/(kq)) under Rα. It can be viewed as a word of length qin the alphabet Ikq.

Let J be the subinterval as before (see section 7.2). We track the Rα′ iterates of J through thewj s as follows:

0. 0-th interval n ∈ [0, klq): Note that any such n can be written as n = mlq + sq + t for someappropriately chosen 0 ≤ m < k, 0 ≤ s < l, 0 ≤ t < q. So (Rα′)

n(J) is a subinterval of thet-th element of wm. So

• t ∈ [0, lq): The Ikq-name agrees with the w0 name repeated l times

• t ∈ [lq, 2lq): The interval crosses the boundary for the Ikq partition and the namechanges to w1 repeated l-times.

. . . . . . .

k − 3 more times to a total of k − 1 times.

. . . . . . .

• t ∈ [(k−1)lq, klq): The interval crosses the boundary for the Ikq partition and the namechanges to wk−1 repeated l-times. Jiklq−1

is geometrically the last subinterval of I1.

Hence the first klq letters of the Ikq-name of any point in J is

wl0wl1w

l2 . . . w

lk−1 (7.3)

1. 1-st interval n ∈ [klq, 2klq):

7.3 Comparison of two periodic processes using a transect - II 30

• t ∈ [klq, (k + 1)lq):

– t ∈ [klq, klq+ q− j1): (Rα′)klq(J) = Jiklq is geometrically the first subinterval of Iq2 .

we use another q − j1 applications of Rα′

to bring J inside Iq0 .

– t ∈ [klq + q − j1, (k + 1)lq − j1): Jiklq+q−j1 is a subinterval of the geometrically first

subinterval of Ikq. In fact it is in the first subinterval of w1. We apply Rα′

(l − 1)qtimes to carry it through l − 1 copies of w1 and it arrives at Iq0

– t ∈ [(k + 1)lq − j1, (k + 1)lq): We apply Rα′

again j1 times to bring it back to Iq1inside w2.

Resulting name is bq−j10 wl−10 ej10 . Here bq−j1j and ej1j are the last q − j1 and the first j1

elements of wj .

• t ∈ [(k + 1)lq, (k + 2)lq): With an identical argument we get the name bq−j11 wl−11 ej11 .

. . . . . . .

k − 3 more times to a total of k − 1 times.

. . . . . . .

• t ∈ [(k + k − 1)lq, 2klq): With an identical argument we get the name bq−j1k−1 wl−1k−1e

j1k−1.

Ji2klq−1is the geometrically last subinterval of I1.

So the klq to 2klq − 1-th letters of the Ikq name of any point in J is given by

bq−j10 w(l−1)0 ej10 b

q−j11 w

(l−1)1 ej11 . . . bq−j1k−1 w

(l−1)k−1 ej1k−1 (7.4)

2. . . .

3. . . .

. . .

. . .

m. m-th interval n ∈ [mklq, (m+ 1)klq): The mklq to (m+ 1)klq − 1-th letters of the Ikq nameof any point in J is given by

bq−jm0 w(l−1)0 ejm0 bq−jm1 w

(l−1)1 ejm1 . . . bq−jmk−1 w

(l−1)k−1 ejmk−1 (7.5)

m+1. . . .

. . .

. . .

q-1. (q-1)-th interval t ∈ [(q − 1)klq, klq2): An identical argument yields the name

bq−jq−1wl−10 ejq−1bq−jq−1wl−1

1 ejq−1 . . . bq−jq−1wl−1k−1e

jq−1 (7.6)

So in conclusion any point in J has a Ikq-name as follows:

w =

q−1∏i=0

k−1∏j=0

(bq−jij wl−1j ejij ) (7.7)

So our periodic process is isomorphic to the symbolic system defined by the above circular operator.

7.4 Back to the regular sequence of periodic processes 31

7.4 Back to the regular sequence of periodic processes

Note that the partition ξn divides T2 into sn identical towers. On each tower , the action is identicaland hence when restricted to a single tower of Iknqn ⊗Isn+1

we get the same analysis as before. Sowe can also copy the previous labeling to a labeling here.

7.5 Symbolic representation of AbC systems

First we describe the general idea behind the representation of the AbC method as a symbolicsystem. We recall some relevant portions from the abstract untwisted AbC method described insection 4. The limit transformation T is obtained as the weak limit of transformations Tn :=Hn Rαn H−1

n . At the n th stage Tn permutes the partition ζn = Hn(ξn) 7.With the above inmind, we recall that (τn, ζn) is a periodic process where τn is the permutation induced by Tn on ζn.

If Q is a partition refined by the levels of the towers of a periodic process τ , then the Q namesof any pointwise realization of τ are constant on the levels of the tower. Hence this is equivalentto naming the levels in the action of τ on various towers. We call the resulting collection of namesthe (τ,Q) names.

Let Q∗ be an arbitrary partition of T2 refined by ζn. We would like to compare the Q∗ names ofpoints under τn and τn+1. We introduce the partition (Hn)−1(Q∗). So the problem of finding the(τn+1,Q∗) name is equivalent to finding the (hn+1 Rαn h−1

n+1,P) names of towers whose levelsconsists of the partition (Hn)−1(ζn+1) = ξn+1.

Tracking movement of individual rectangles

For notational simplicity we put kn = k, qn = q, Ii = Iqni and Ji = Iqn+1

i . Fix R ∈ ξn+1.Assume that Rn+1

i,j = h−1n+1(R) ⊂ Ji for some i and Ii is not the last subinterval of an interval

in the partition Ikq. In this case both Rαn(Rn+1i,j ) and Rαn+1(Rn+1

i,j ) belong to the same element of

Ikq × T1.Since Rαn commutes with hn+1 and hn+1 permutes the atoms of Iknqn ⊗Isn+1

, we have hn+1 Rαn+1 h−1

n+1(R) = hn+1 Rαn+1(Rn+1i,j ) and hn+1 Rαn h−1

n+1(R) = Rαn(R) belong to the sameatom of Iknqn ⊗ Isn+1 . So they share the same P-name.

On the other hand if Ji is the last subinterval of an interval in the partition Ikq, then Rαn+1

sends Rn+1i,j to the geometrically first subrectangle of a new element R′ of Ikq ⊗ Isn+1

. Thus

hn+1 Rαn+1 h−1n+1(R) is a subrectangle of hn+1(R′).

Tracking movement of levels through a tower

We note that the base of the towers of (τn+1, ζn+1) are the rectangles Hn+1(Rn+10,j )sj=0 while those

for the towers of (hn+1 Rαn+1 h−1n+1,P) are hn+1(Rn+1

0,j )sj=0.

So with F0 := hn+1(Rn+10,j ) as the base, we define Ft := (hn+1 Rαn+1 h−1

n+1)t(hn+1(Rn+10,j )) =

(hn+1 (Rαn+1)t)(Rn+10,j )) = hn+1(Rn+1

it,j) where it is the dynamical ordering of the t-th interval

under iterations by Rαn+1 .We recall the labeling we used in section 7.3 using the w, b and e s. So if t is such that Jit

is labeled with a part of w, the two transformations hn+1 Rαn+1 hn+1 and Rαn move Ft to asubrectangle of the same element of Ikq ⊗ Isn+1

and hence the same element of P.

7 So ζn is just ξn ordered in a different way.

7.5 Symbolic representation of AbC systems 32

For j < k, t < q and s < sn+1, define:

Rj,t,s :=[j + tk

kq,j + tk + 1

kq

)×[ s

sn+1,s+ 1

sn+1

)(7.8)

And uj,s be the sequence of P-names for

hn+1(Rj,t,s)q−1t=0 (7.9)

The word uj,s is the sequence of P-names for the levels of the tower (Rαn)t hn+1(Rn+1i,s )qt=0,

fr any Ji ⊂ [j/(kq), (j + 1)/(kq)).Using transects, we now describe the P-name of the orbit of F0 under hn+1 Rαn+1 h−1

n+1. We

use t to denote the iterations of hn+1 Rαn+1 h−1n+1 and hence it also denotes the level of the tower.

0. 0-th interval t ∈ [0, klq):

• t ∈ [0, lq): The (hn+1 Rαn+1 h−1n+1,P) name agrees with the Rαn -name u0,s repeated

l-times.

• t ∈ [lq, 2lq): The interval crosses the boundary for the Ikq partition and the namechanges to u1,s repeated l-times.

. . . . . . .

k − 3 more times to a total of k times.

. . . . . . .

• t ∈ [(k−1)lq, klq): The interval crosses the boundary for the Ikq partition and the namechanges to uk−1,s repeated l-times. Jiklq−1

is geometrically the last subinterval of I1.

1. 1-st interval t ∈ [klq, 2klq):

• t ∈ [klq, (k + 1)lq):

– t ∈ [klq, klq + q − j1): Jiklq is geometrically the first subinterval of I2. This portionof the transect is labeled with q − j1 copies of b.

– t ∈ [klq + q − j1, (k + 1)lq − j1): Jiklq+q−j1 is a subinterval of the geometrically firstsubinterval of Ikq. So Ft is the first letter of u0,s. For the next q(l − 1) iterationsthe hn+1 Rαn+1 h−1

n+1 names are the same as the Rαn names and obtain the nameu0,s.

– t ∈ [(k + 1)lq − j1, (k + 1)lq): This segment of the transects is labeled by j1 copiesof e.

Resulting name is bq−j1ul−10,s e

j1 .

• t ∈ [(k + 1)lq, (k + 2)lq): With an identical argument we get the name bq−j1ul−11,s e

j1 .

. . . . . . .

k − 3 more times to a total of k times.

. . . . . . .

• t ∈ [(k + k − 1)lq, 2klq): With an identical argument we get the name bq−j1ul−1k−1,se

j1 .Ji2klq−1

is the geometrically last subinterval of I1.

7.5 Symbolic representation of AbC systems 33

So the name of the 1-st interval is:

bq−j1ul−10,s e

j1bq−j1ul−11,s e

j1 . . . bq−j1ul−1k−1,se

j1 (7.10)

2. . . .

3. . . .

. . .

. . .

m. m-th interval t ∈ [mklq, (m+ 1)klq): An identical argument yields the name

bq−jmul−10,s e

jmbq−jmul−11,s e

jm . . . bq−jmul−1k−1,se

jm (7.11)

m+1. . . .

. . .

. . .

q-1. (q-1)-th interval t ∈ [(q − 1)klq, klq2): An identical argument yields the name

bq−jq−1ul−10,s e

jq−1bq−jq−1ul−11,s e

jq−1 . . . bq−jq−1ul−1k−1,se

jq−1 (7.12)

Theorem 7.13. Let F0 := hn+1(R0,s) be the base of a tower T for hn+1 Rαn+1 h−1n+1. Then the

P-names of T agree with

u :=

q−1∏i=0

k−1∏j=0

(bq−jiul−1j,s e

ji) (7.14)

on the interior of u.

Corollary 7.15. With Ji ⊆ [j/(kq), (j + 1)/kq − (1/qn+1)), j < kq and R = Rni,s, the levels

(Rαn)t(R)q−1t=0 coincide with the levels (hn+1 Rαn+1 h−1

n+1)thn+1(R)q−1t=0 in the tower for τn.

In particular their Q∗ names agree.

Corollary 7.16. For a set x ∈ X having measure at least 1−3/ln, the (τn,Q∗) and (τn,Q∗) namesof x agree on the interval [−q, q].

Corollary 7.17. Suppose that x ∈ Γn and x is on level tn of a τn-tower and the level tn+1 of aτn+1-tower. Let wn be the Q∗-name of x with respect to τn and wn+1 be the Q∗-name of x withrespect to τn+1. Then wn+1|[tn+1−tn,tn+1+qn−tn) = wn.

7.5 Symbolic representation of AbC systems 34

The symbolic representation

We define the following two sets:

B := x ∈ T2 : for some m ≤ n, x ∈ Γn and H−1m (x) ∈ Bm (7.18)

E := x ∈ T2 : for some m ≤ n, x ∈ Γn and H−1m (x) ∈ Em (7.19)

We define the partition

Q := Ai : i < s0 ∪ B,E (7.20)

where Ais0i=0 is the partition ζ0|T2\B∪E . Explicitly Ai := H0([0, 1)× [s/s0, (s+ 1)/s0) \B ∪ E).

We will construct (T,Q)-names for each x ∈ T2 using the alphabet Σ∪b, e where Σ := ais0−1i=0 .

So the the name of point x ∈ T2 will be an f ∈ (Σ ∪ b, e)Z with f(n) = ai ⇐⇒ Tn(x) ∈ Ai,f(n) = b ⇐⇒ Tn(x) ∈ B and f(n) = e ⇐⇒ Tn(x) ∈ E.

We use induction for a complete description of the (T,Q)-names of points in ∪Γn. Let F0 :=Hn+1(Rn+1

0,s∗ ) for some s∗ < qn+1 be the base of a tower T of τn+1.Inductively we assume that the (τn,Q)-names of the towers with bases Hn(Rn0,s)

sns=0 are

u0, . . . , usn−1.At the n+ 1-th stage of the induction we start by defining words w0, . . . wkn−1 by setting

wj = us ⇐⇒ hn+1

([ j

knqn,j + 1

knqn

)×[ s∗

sn+1,s∗ + 1

sn+1

))⊆ Rn0,s (7.21)

We say that (w0, . . . , wkn−1) is the sequence of n-words associated with T .

Definition 7.22. We define a circular system by inductively specifying the sequence Wnn∈N.We put W0 := ais0i=0. After having defined Wn we define

Wn+1 := Cn+1(w0, . . . , wkn−1) : (w0, . . . , wkn−1) is associated with a tower T in τn+1 (7.23)

We say that Wnn∈N is the construction sequence associated with the AbC construction.

Theorem 7.24. Suppose T is an AbC transformation satisfying Requirements 1, 2, 3 and the lnparameters are assumed to grow fast enough. Let Wnn∈N be the construction sequence associatedwith T and let K be its circular system.

The almost all x ∈ T2 have Q-names in K. In particular there is a measure ν on K that makes(K,B, ν, sh) isomorphic to the factor of T2 generated by Q.

Proof. Let M be any natural number. Then for a.e. x ∈ T2, there exists an N = N(x) such thatfor all n > N the first or last M levels of τn does not contain x.

Let x ∈ Γn0⊂ ∪nΓn be an arbitrary point. Then x belongs to a tower T of τn and we may in

light of the previous paragraph assume that x does not belong to the first of last M levels of T .Then we can use corollary 7.17 to conclude that the if x is in the tn-th level of a τn-tower, then theT -name of x agrees with the τn name of x on the interval [−tn, qn − tn).

Thus we can apply theorem 7.13 to conclude that with P = (Hn)−1(Q), we see that a tower Tfor τn has the name:

w =

qn−1−1∏i=0

kn−1−1∏j=0

(bqn−jiul−1j eji) (7.25)

7.5 Symbolic representation of AbC systems 35

where wjkn−1−1j=0 is the sequence of words associated with T . For x in the tn-th level of T , the

Q-name of x on the interval [−tn, qn − tn) is

qn−1∏i=0

kn−1∏j=0

(bqn−jiul−1j eji) = Cn(w0, . . . , wkn−1−1) (7.26)

Since M < min(tn, qn − tn) we have x|[−M,M ] is a subword of Wn.It follows that the factor of (T2,B, µ, T ) corresponding to the partition Q is a factor of the

uniform circular system defined by the constructions sequence Wnn∈N.Conversely, by lemma 3.6, the uniform circular system K determined by Wnn∈N is character-

ized as the smallest shift invariant closed set intersecting every basic open set C0(w) in (Σ∪b, e)Zdetermined by some w ∈ Wn. However each w ∈ Wn is represented on Γn ∩ T for some T , henceeach C0(w) has non empty intersection with the set of words arising from (T,Q)-names.

Lemma 7.27. Let a0, b0 ∈ N. Then for a.e. x and large enough n, x ∈ Γn, x does not occur inthe first a0 levels or last b0 levels f a tower of τn. In particular for a.e. x ∈ X there are a > a0,b > b0 such that the Q-name of x restricted to the interval [−a, b) belongs to Wn.

Corollary 7.28. For a.e. x ∈ T2 the Q-name of x is in S. In particular if ν is the unique non-atomic shift-invariant measure on K, then the factor of (T2,B, µ, T ) generated by Q is isomorphicto (K,B, sh, ν).

Lemma 7.29. Suppose that for all n, the map sending a tower T for τn to the sequence of Q-namesassociated to T is a one-one function. Then Q generates the transformation T .

Proof. Without loss of generality assume that H0 is the identity map. Since Hnξnn∈N is adecreasing and generating sequence of partitions, and µ(Γn) 1, we conclude that Hnξn∩Γnn=Nalso generate B. So we will be done once we show that Hnξn∩Γn belongs to the smallest translationinvariant σ-algebra generated by ∨Nn=−NT

i(Q∪ B,E)N∈N.Each P ∈ Hnξn is the t-th level of some tower T for τn. Let w ∈ (Σ ∪ b, e)qn be the (τn,Q)-

name of T . So the j-th letter of w determines and Sji ∈ (B ∪ E ∪ Ai : i < s0). Since(τn,Q)-name of T is correct on Γn,

P ∩ Γn ⊆qn−1⋂j=0

T t−j(Sij ) ∩ Γn (7.30)

On the other hand, since the map sending towers to names w is one to one, we see

P ∩ Γn ⊇qn−1⋂j=0

T t−j(Sij ) ∩ Γn (7.31)

This completes the proof.

Lemma 7.32. If the AbC construction is done satisfying the Requirements 1, 2 and 3 then Qgenerates the transformation T .

7.5 Symbolic representation of AbC systems 36

Proof. We will show that the Q-names associated with any two τn-towers T and T are different.For n = 0 it is clear by definition. We assume that this is true upto n and we show it

for n + 1. Let Rn+10,s and Rn+1

0,s′ be the bases of T and T ′ respectively. Requirement 3 impliesthat the kn-tuples (j0, . . . , jkn−1) and (j′0, . . . , j

′kn−1) associated with s and s′ are distinct. Let

wt be the names of the towers of τn with bases Rn0,s. Then all the wt s are distinct as perour induction hypothesis. Also by the Theorem 7.13, the Q-names of T and T ′ are given byC(wj0 , . . . , wjkn−1

) and C(wj′0 , . . . , wj′kn−1) respectively. Since (j0, . . . , jkn−1) and (j′0, . . . , j

′kn−1) are

different, C(wj0 , . . . , wjkn−1) and C(wj′0 , . . . , wj′kn−1

) are different.

We conclude the proof using lemma 7.32.

Theorem 7.33. Suppose the measure preserving system (T2,B, µ, T ) is built by the AbC methodusing fast growing coefficients and hn s in the scheme satisfies requirements 1 to 3. Let Q bethe partition defined earlier. Then the Q-names describe a strongly uniform circular constructionsequence Wnn∈N. Let K be the associated circular system and φ : T2 → K be the map sendingeach x ∈ T2 to its Q-name. Then φ is one to one on a set of µ measure one. Moreover, there isa unique non-atomic shift invariant measure ν concentrating on the range of φ and this measure isergodic. In particular (T2,B, µ, T ) is isomorphic to (K,B, ν, sh).

Proof. With fast growing ln parameters, we know that the sequence Wn forms a uniform circularsequence and hence there is a unique shift invariant non-atomic ergodic measure ν on S ⊂ K. Usinglemma 7.27, we have that the range of φ is a subset of the set S. So the factor determined by φ isisomorphic to (S,B, µ, sh)

Since requirements 1,2 and 3 are satisfied, the partition Q generates T2 and hence φ is anisomorphism.

Theorem G. Consider three sequence of natural numbers knn∈N, lnn∈N, snn∈N tending toinfinity. Assume that

1. ln grows sufficiently fast.

2. sn divides both kn and sn+1.

Let Wnn∈N be a circular construction sequence in an alphabet Σ ∪ b, e such that

1. W0 = Σ, |Wn+1| = sn+1 for any n ≥ 1.

2. (Uniform) For each w′ ∈ Wn+1, and w ∈ Wn, if w′ = C(w0, . . . , wKn−1), then here are kn/snmany j with w = wj.

Then,

1. Wnn∈N is a uniform construction sequence. If K is the associated symbolic shift then thereis a unique non atomic ergodic measure ν on K.

2. There is a real-analytic measure preserving transformation T defined on T2 such that thesystem (T2,B, µ, T ) is isomorphic to (K,B, ν, sh).

Proof. Suppose that we have defined hmnm=0 in the AbC process. From the definition of uniformcircular systems (see 3.18), we can find Pn+1 ⊆ (Wn)kn such that Wn+1 is the collection of w′ suchthat for some sequence (w0, . . . , wkn−1) ∈ Pn+1, w

′ = C(w0, . . . , wkn−1). We enumerate Pn+1 =w′0, . . . , w′sn+1−1. We apply lemma 4.11 to get hn+1 from w′0, . . . , w

′sn+1−1.

Proof of Theorem E 37

With hnn∈N constructed as above, we note that requirement 1 is satisfied because sn ∞,requirement 2 and 3 follows from the fact that the words in Pn+1 are distinct.

We note that the sequence Pnn∈N determines hn in the AbC method which in turn determinesa neighborhood in the real-analytic topology in which the resulting AbC diffeomorphism belongs.

Conversely, different choices of Pn gives distant hn’s and hence distant h(a)n in the analytic topology.

We record this as this is going to be crucial in the proof of an anti-classification result.

Proposition 7.34. Suppose Unn ∈ N and Wnn ∈ N are two construction sequences for circularsystems and M is such that Un =Wn ∀n ≤M . If S and T are the real-analytic realizations of thecircular systems using the AbC method given in this paper, then

dρ(S, T ) < εM (7.35)

Proof. The sequences kUn , lUn , hUn , sUnn∈N and kWn , lWn , hWn , sWn n∈N associated with the two con-struction sequences determining approximations Snn∈N and Tnn∈N to diffeomorphisms S andT has the property kUn = kWn , l

Un = lWn , h

Un = hWn , s

Un = sWn for all n ≤ M . So SM = TM . It follows

from equations 6.5 and 6.6 that

dρ(Sm, S) < εM/2 (7.36)

dρ(Tm, T ) < εM/2 (7.37)

Combining the two together with the triangle inequality we get the result.

8 Proof of Theorem E

In this Section we survey the key steps from [FRW11] and [FW3pp] and adapt them to our proofof Theorem E in the real-analytic category.

8.1 Trees, groups and equivalence relations

During the construction the following maps will prove useful.

Definition 8.1. We define a map M : T rees → NN by setting M (T ) (s) = n if and only if n isthe least number such that σn ∈ T and |σn| = s. Dually, we also define a map s : T rees→ NN bysetting s (T ) (n) to be the length of the longest sequence σm ∈ T with m ≤ n.

Remark. When T is clear from the context we write M(s) and s(n). We also note that s(n) ≤ nand that s as well as M are continuous functions when we endow N with the discrete topology andNN with the product topology.

Related to the structure of the tree we will specify several equivalence relations Qns on the wordsWn. For this purpose, we recall some general notions on equivalence relations.

Definition 8.2. Let X be a set and Q as well as R equivalence relations on X.

• We write R ⊆ Q and say that R refines Q if considered as sets of ordered pairs we haveR ⊆ Q.

8.1 Trees, groups and equivalence relations 38

• We define the product equivalence relation Qn on Xn by setting x0x1 . . . xn−1 ∼ x′0x′1 . . . x′n−1

if and only if xi ∼ x′i for all i = 0, 1, . . . , n− 1.

In [FRW11] groups associated to trees are used to approximate conjugacies. These groups aredirect sums of Z2 = Z/2Z and are called groups of involutions.

If G =∑i∈I (Z2)i and B = ri : i ∈ I is a distinguished basis, then we call the elements ri ∈ B

generators and we have a well-defined notion of parity for elements in G: an element g ∈ G iscalled even if it can be written as the sum of an even number of elements in B. Otherwise, itis called odd. Parity is preserved under homomorphisms sending the distinguished basis of onegroup to the distinguished basis of the other. This also yields that for an inverse limit system ofgroups of involutions Gs : s ∈ N (where each group has a distinguished set of generators) withhomomorphisms ρt,s : Gt → Gs for s < t, the elements of the inverse limit lim←−Gs have a well definedparity.

For a tree T ⊂ NN we assign to each level s a group of involutions Gs (T ) by taking a sum ofcopies of Z2 indexed by the nodes of T at level s. Moreover, we view these nodes of T at level s asthe distinguished generators of

Gs (T ) =∑

τ∈T , |τ |=s

(Z2)τ .

For levels s < t of T we have a canonical homomorphism ρt,s : Gt (T ) → Gs (T ) that sends agenerator τ of Gt (T ) to the unique generator σ of Gs (T ) that is an initial segment of τ . Wedenote the inverse limit of 〈Gs (T ) , ρt,s : s < t〉 by G∞ (T ) and we let ρs : G∞ (T ) → Gs (T ) bethe projection map.

Since there is a one-to-one correspondence between the infinite branches of T and infinite se-quences (gs)s∈N of generators gs ∈ Gs (T ) with ρt,s (gt) = gs for t > s, we obtain the followingcharacterization.

Fact 8.3. Let T ⊂ NN be a tree. Then

1. G∞ (T ) has a nonidentity element of odd parity if and only if T is ill-founded (i.e. has aninfinite branch).

2. G∞ (T ) has a nonidentity element of even parity if and only if T has at least two infinitebranches).

In order to make the elements of G∞ (T ) to correspond to conjugacies, one builds symmetriesinto our construction using the following finite approximations to G∞ (T ). We let Gn0 (T ) be thetrivial group and for s > 0 we let

Gns (T ) =∑

(Z2)τ , where the sum is taken over τ ∈ T ∩ σm : m ≤ n , |τ | = s.

When T is clear from the context, we will frequently write Gns .During the course of construction we will consider group actions of Gns on our quotient spaces

Wn/Qns . Here, we will need to control systems of such group actions on the refining equivalencerelations. For that purpose, the following general definitions will prove useful.

Definition 8.4. Suppose

• Q and R are equivalence relations on a set X with R refining Q,

8.2 Specifications and timing assumptions 39

• G and H are groups with G acting on X/Q and H acting on X/R,

• ρ : H → G is a homomorphism.

Then we say that the H action is subordinate to the G action if for all x ∈ X, whenever [x]R ⊂ [x]Qwe have h[x]R ⊂ ρ(h)[x]Q.

Definition 8.5. If G acts on X, then the canonical diagonal action of G on Xn is defined by

g (x0x1 . . . xn−1) = gx0 gx1 . . . gxn−1.

If G is a group of involutions with a distinguished collection of free generators, then we define theskew diagonal action on Xn by setting

g (x0x1 . . . xn−1) = gxn−1 gxn−2 . . . gx0

for any canonical generator g.

Remark. We stress that the skew diagonal actions by elements of G of odd parity reverse the ordersof the xi’s while group elements of even parity preserve the order.

Remark. Recalling the notion of a product equivalence relation from Definition 8.2 we can identifyXn/Qn with (X/Q)

nin an obvious way. Then we can also extend an action of G on X/Q to the

diagonal or skew diagonal actions on (X/Q)n

in a straightforward way.

8.2 Specifications and timing assumptions

In [FRW11] the sequences (Wn)n∈N, equivalence relations Qns , and group actions of Gns satisfyingseveral specifications are constructed inductively. For each n ∈ N with σn ∈ T one constructsa set of words Wn = Wn(T ) in the basic alphabet 0, 1 and the construction depends only onT ∩ σm | m ≤ n. To start we set W0 = 0, 1.

(E1) All words in Wn have the same length hn and |Wn| := sn is a power of 2.

(E2) If σm and σn are consecutive elements of T , then every word in Wn is built by concatenatingwords in Wm. Every word in Wm occurs in each word of Wn exactly p2

n many times. Here, pnis a large prime number chosen when σn is considered.

(E3) If σm and σn are consecutive elements of T , w ∈ Wn and w = pw1 . . . wls, where each wi ∈ Wmand p and s are strings in 0, 1 of length less than the length of the m-words, then both pand s are the empty words.

If w,w′ ∈ Wn, then the first half of w′ is not equal to w, i.e. if w = w1 . . . whn/hm and

w′ = w′1 . . . w′hn/hm

, where wi, w′i ∈ Wm, and k = b hn/hm2 c + 1, we have wk . . . whn/hm 6=

w′1 . . . w′hn/hm−k−1.

Remark. In particular, these specifications say that Wn | σn ∈ T is a uniquely readable andstrongly uniform construction sequence.

In the next step, we specify equivalence relations Qns on Wn which are defined for s ≤ s(n). Forthis purpose, we take a decreasing summable sequence (εn)n∈N and a strictly increasing sequencee(n) of positive integers. To start, we let Q0

0 be the equivalence relation on W0 = 0, 1 which hasone equivalence class, i.e. both elements of 0, 1 are equivalent.

8.2 Specifications and timing assumptions 40

(Q4) Suppose that n = M(s). Then any two words in the same Qns class agree on an initial segmentof length at least (1− εn) hn.

(Q5) For n ≥ M(s) + 1 we can consider words in Wn as concatenation of words from WM(s) and

define Qns as the product equivalence relation of QM(s)s .

(Q6) Qns+1 refines Qns and each Qns class contains 2e(n) many Qns+1 classes.

Remark. Each equivalence relation Qns will induce an equivalence relation on rev(Wn), which wewill also call Qns , as follows: rev(w), rev(w′) ∈ rev(Wn) are equivalent with respect to Qns if andonly if w,w′ ∈ Wn are equivalent with respect to Qns .

Remark. By (Q5) we can view Wn/Qns as sequence of elements WM(s)/QM(s)s and similarly for

rev(Wn)/Qns . It also follows that Qn0 is the equivalence relation on Wn which has one equivalenceclass.

Remark. We denote the number of Qns equivalence classes by Qns and the cardinality of each classby Cns . In case that the exponent is not relevant we will refer to the Qns as Qs. For u ∈ Wn we write[u]s for its Qns class.

We now list specifications on the group actions.

(A7) Gns acts freely on Wn/Qns ∪ rev(Wn/Qns ) and the Gns action is subordinate to the Gns−1 actionvia the natural homomorphism ρs,s−1 : Gns → Gns−1.

(A8) The canonical generator ofGM(s)s sends elements of WM(s)/Q

M(s)s to elements of rev(WM(s))/Q

M(s)s

and vice versa.

(A9) Suppose M(s) < n, σm and σn are consecutive elements of T and we view Gns = Gms ⊕ H.Then the action of Gms on Wm/Qms ∪ rev(Wm/Qms ) is extended to an action on action onWn/Qns ∪ rev(Wn/Qns ) by the skew diagonal action. If H is nontrivial, then its canonicalgenerator maps Wn/Qns to rev(Wn/Qns ).

All these specifications build certain symmetries into the words that allow nodes of the tree togive increasingly precise information about invertible graph joinings. The intent of the followingspecifications is to give a mechanism for showing that any joining not arising from branches throughthe tree are independent joinings over a non-trivial factor, i.e. they cannot give an isomorphism.

(J10) Suppose σm and σn are consecutive elements of T . Let u and v be elements of Wn ∪ rev(Wn)and let 1 ≤ t < (1− εn) hn

hm. Let j0 be a number between εn

hnhm

and hnhm− t. Then for each pair

u′, v′ ∈ Wm ∪ rev(Wm) such that u′ has the same parity as u and v′ has the same parity as v,let r(u′, v′) be the number of j < j0 such that (u′, v′) occurs in (shthm(u), v) in the (jhm)-thposition in their overlap. Then ∣∣∣∣r(u′, v′)j0

− 1

s20

∣∣∣∣ < εn.

(J11) Suppose σm and σn are consecutive elements of T . Let u ∈ Wn and v ∈ Wn ∪ rev(Wn). Welet s = s(u, v) be the maximal i such that there is a g ∈ Gmi such that g[u]i = [v]i. Letg = g(u, v) be the unique g with this property and (u′, v′) ∈ Wm × (Wm ∪ rev(Wm)) such thatg[u′]s = [v′]s. Let r(u′, v′) be the number of occurrences of (u′, v′) in (u, v). Then:∣∣∣∣∣r(u′, v′)hn/hm

− 1

Qns

(1

Cns

)2∣∣∣∣∣ < εn.

8.3 The continuous function F : T rees→MPT 41

To satisfy so-called timing assumptions in [FW3pp, Section 7.2.1] for the corresponding circularsystem one also needs this strengthening of a special case.

(J11.1) Suppose σm and σn are consecutive elements of T . Let u ∈ Wn, v ∈ Wn ∪ rev(Wn), and[u]1 /∈ Gn1 [v]1. Let j0 be a number between εn

hnhm

and hnhm

. Suppose that I is either an initialor a tail segment of the interval [0, hn − 1] ∩ Z having length j0hm. Then for every pair(u′, v′) ∈ Wm × (Wm ∪ rev(Wm)) such that u′ has the same parity as u and v′ has the sameparity as v, let r(u′, v′) be the number of occurrences of (u′, v′) in (u I, v I). Then:∣∣∣∣r(u′, v′)j0

− 1

s2n

∣∣∣∣ < εn.

Under these specifications and the timing assumptions the following existence result for syn-chronous and anti-synchronous isomorphisms is shown in [FW3pp, Theorem 92].

Fact 8.6. Suppose Kc is a circular system satisfying the timing assumptions and view it as anelement T of MPT. Then:

1. If there is an isomorphism φ : Kc → Kc such that φ /∈ Tn | n ∈ Z, then there is an isomor-phism ψ : Kc → Kc such that φ /∈ Tn | n ∈ Z and ψπ is the identity map.

2. If there is an isomorphism φ : Kc → (Kc)−1, then there is an isomorphism ψ : Kc → (Kc)−1

such that ψπ = \.

8.3 The continuous function F : T rees→MPT

The specifications allow to construct the following continuous map F : T rees → MPT, which isthe main result of [FRW11].

Proposition 8.7. There is a continuous one-to-one map F : T rees → MPT such that for T ∈T rees, if T = F s(T ):

1. T has an infinite branch if and only if T ∼= T−1

2. T has two distinct infinite branches if and only if

C(T ) 6= Tn | n ∈ Z.

More specifically, the following facts for the map F hold true.

Fact 8.8. The transformations in the range of F are strongly uniform odometer based transfor-mations and for S in the range of F we have S ∼= S−1 if and only if there is an anti-synchronousisomorphism between S and S−1.

Fact 8.9. If S is in the range of F and C(S) 6= Sn | n ∈ Z, then there is a synchronous isomor-phism φ ∈ C(S) \ Sn | n ∈ Z such that for some n ∈ N there is a non-identity element g ∈ Gn1satisfying for all generic s ∈ K and all large enough m ∈ N that if u and v are the principalm-subwords of s and φ(s), respectively, then [v]1 = g[u]1.

Fact 8.10. For every N ∈ N there is M ∈ N such that for T , T ′ ∈ T rees with T ∩σn | n ≤M =T ′ ∩ σn | n ≤M the first N steps of the construction sequence for F (T ) are equal to the first Nsteps of the construction sequence for F (T ′), i.e. (Wk(T ))k<N = (Wk(T ′))k<N .

8.4 Numerical requirements 42

8.4 Numerical requirements

To construct the map F : T rees → MPT one builds a construction sequence (Wn (T ))n∈N sat-isfying our specifications for each T ∈ T rees. Moreover, this has to be done in such a way thatWn (T ) is entirely determined by T ∩ σm : m ≤ n. Therefore, in [FRW11] the construction isorganized as follows: For each n and for each subtree S ⊆ σm : m ≤ n and each σm ∈ S we buildWm (S). By induction we want to pass from stage n − 1 to stage n. So, we assume that we haveconstructed (Wm (S))σm∈S for each subtree S ⊆ σm : m ≤ n− 1. In the inductive step, we nowhave to construct (Wm (S))σm∈S for each subtree S ⊆ σm : m ≤ n.

Obviously, for those trees S with σn /∈ S there is nothing to do. So, one has to work on thefinitely many trees S with σn ∈ S. We list those in arbitrary manner as S1, . . . ,SE. Assume thatT is the e-th tree on this list and that we have constructed the collections Wn (Si) of n-words forall i ∈ 1, . . . , e− 1. Let P be the collection of prime numbers occurring in the prime factorizationof any of the lengths of the words that we have constructed so far. Then one wants to constructWn (T ), Qns (T ), and Gns (T ). For that purpose, one uses the induction assumption that for mbeing the largest number less than n such that σm ∈ T we have Wm = Wm (T ), Qms = Qms (T ),and Gms = Gms (T ) satisfying our specifications.

During the course of the construction several parameters with numerical conditions about growthand decay rates appear. In [FW3pp, Section 10] all their recursive requirements and interdepen-dencies are listed and their consistency is resolved. In particular, at stage n the parameter ln ischosen last. Hence, we can choose it sufficiently large to guarantee the convergence to a real-analyticdiffeomorphism in our Theorem G.

8.5 F s = R F F is a continuous reduction

Finally, we are ready to prove Theorem E. Therefor, we recall that Theorem G gives to every stronglyuniform circular construction sequence Wnn∈N with |Wn| = sn → ∞ and circular coefficients(kn, ln)n∈N, where (ln)n∈N grows sufficiently fast, a diffeomorphism T ∈ Diff ω

ρ (T2, µ) measuretheoretically isomorphic to the circular system Kc. Hereby, we obtain a map from circular systemswith fast growing coefficients to Diff ω

ρ (T2, µ). We denote this map with our choice of the sequence(ln)n∈N from the previous Subsection by R and point out that it preserves isomorphism.

Hereby, we define the map F s = R F F : T rees → Diff ωρ (T2, µ) and show that it is a

continuous reduction.

Lemma 8.11. F s : T rees→ Diff ωρ (T2, µ) is a continuous function.

Proof. Let T = F s(T ) for T ∈ T rees and U be an open neighborhood of T in Diff ωρ (T2, µ). Let

T c = F F (T ) be the circular system such that R(T c) = T . By Proposition 7.34 there is M ∈ Nsufficiently large such that for all S ∈ Diff ω

ρ (T2, µ) in the range of R we have the following property:If (Wn(T c))n≤M = (Wn(Sc))n≤M , then S ∈ U . Here, Sc denotes the circular system such thatS = R(Sc). Moreover, (Wn(T c))n∈N and (Wn(Sc))n∈N denote the construction sequences of T c andSc, respectively. We recall from Subsection 3.4 that (Wn(T c))n≤M is determined by the first M +1members in the construction sequence of the odometer based system F (T ), i.e. it is determinedby (Wn(T ))n≤M . By Fact 8.10 there is a basic open set V ⊆ T rees containing T such that for allS ∈ T rees the first M + 1 members of the construction sequences of F (T ) and F (S) are the same,i.e. (Wn(T ))n≤M = (Wn(S))n≤M . Then it follows that F s(S) ∈ U for all S ∈ V .

To conclude the proof of Theorem E we show that F s is a reduction.

Further applications 43

Proof of Theorem E. To see that F s is a reduction, it suffices to check that F F is a reductionbecause R preserves isomorphism. This follows from [FW3pp, Section 8.1] and we only present theproof of the first part of the Theorem. Here, we let T = F s(T ), K = F (T ), and Kc = F F (T ) forT ∈ T rees.

Suppose T ∈ T rees has an infinite branch. By Proposition 8.7 and Fact 8.8 there is an anti-synchronous isomorphism φ : K→ K−1. Then we can apply Fact 3.24 on the functor F and obtainthat there is an anti-synchronous isomorphism φc : Kc → (Kc)−1.

To show the converse direction we suppose that T ∼= T−1, which yields Kc ∼= (Kc)−1. Since theconstruction sequence (Wn)n∈N for Kc satisfies the timing assumptions, Fact 8.6 shows that thereis an anti-synchronous isomorphism φc : Kc → (Kc)−1. Hence, we can apply Fact 3.24 again andobtain that there is an anti-synchronous isomorphism φ : K → K−1. By Proposition 8.7 T has aninfinite branch.

9 Further applications

9.1 An uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomorphisms

We recall the definition of the f distance between strings of symbols introduced by Feldman [Fe76](a zero entropy version of this property was introduced independently by Katok [Ka77]).

Definition 9.1. A match between two strings of symbols a1a2 . . . an and b1b2 . . . bm from a givenalphabet Σ, is a collection I of pairs of indices (is, js), s = 1, . . . , r such that 1 ≤ i1 < i2 < · · · <ir ≤ n, 1 ≤ j1 < j2 < · · · < jr ≤ m and ais = bis for s = 1, 2, . . . , r. Then

f(a1a2 . . . an, b1b2 . . . bm) = 1− 2 sup|I| : I is a match between a1a2 · · · an and b1b2 · · · bmn+m

.

We will refer to f(a1a2 · · · an, b1b2 · · · bm) as the “f -distance” between a1a2 · · · an and b1b2 · · · bm,even though f does not satisfy the triangle inequality unless the strings are all of the same length.A match I is called a best possible match if it realizes the supremum in the definition of f .

In the case of zero entropy, the f distance allows a simple definition of the loosely Bernoulliproperty.

Definition 9.2 (Loosely Bernoulli in the case of zero entropy). A measure-preserving process(T,P, ν) is zero-entropy loosely Bernoulli if for every ε > 0, there exists a positive integer K = K(ε)and a collection G of “good” atoms of ∨K1 T−iP with total measure greater than 1− ε such that foreach pair A,B of atoms in G, fK(x, y) < ε for x ∈ A, y ∈ B.

Remark. If this condition is satisfied, then routine estimates show that the (T,P, ν) process indeedhas zero entropy. Sometimes, zero-entropy loosely Bernoulli transformations are called standard orloosely Kronecker.

The following well-known fact from [Fe76] and [ORW82] as stated in [GKpp, Property 2.4] willprove useful in our arguments.

Fact 9.3. Suppose a and b are strings of symbols of length n and m, respectively, from an alphabetΣ. If a and b are strings of symbols obtained by deleting at most bγ(n + m)c terms from a and baltogether, where 0 < γ < 1, then

f(a, b) ≥ f(a, b)− 2γ. (9.4)

9.1 An uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomorphisms 44

Moreover, if there exists a best possible match between a and b such that no term that is deletedfrom a and b to form a and b is matched with a non-deleted term, then

f(a, b) ≥ f(a, b)− γ. (9.5)

Likewise, if a and b are obtained by adding at most bγ(n+m)c symbols to a and b, then (9.5) holds.

Using the notation from Section 3.4 we define the (n+ 1)-blocks w(n+1)j , j = 1, . . . , sn+1, in the

odometer-based system as follows:

w(n+1)j =

((w

(n)1

)(ln−1)2j−1 (w

(n)2

)(ln−1)2j−1

. . .(w(n)sn

)(ln−1)2j−1)(ln−1)2·(sn+1−j)+1

.

Applying the circular operator Cn yields the following (n+ 1)-blocks in the corresponding circularsystem:

qn−1∏i=0

(sn∏t=1

(bqn−ji

(cn

(w

(n)t

))ln−1

eji)(ln−1)2j−1)(ln−1)2·(sn+1−j)+1

for j = 1, . . . , sn+1.

Since the newly introduced spacers occupy a proportion of at most 2ln

in substantial substrings of

these blocks with at least qn+1

qnconsecutive symbols, we will ignore them in the following consider-

ation. This might decrease the f distance between substantial substrings by at most 4ln

accordingto the aforementioned Fact 9.3. After ignoring the new spacers, the (n + 1)-blocks in the circularsystem look as follows for j = 1, . . . , sn+1:((

cn

(w

(n)1

))(ln−1)2j (cn

(w

(n)2

))(ln−1)2j

. . .(cn

(w(n)sn

))(ln−1)2j)qn(ln−1)2(sn+1−j)+1

.

Apparently, they are of the same form as the patterns in [Fe76]. As already observed in [ORW82],the explicit number of blocks and lengths of cycles do not matter for the sake of the argument. Itis just important that the number sn+1 of types tends to infinity and that the ratio of the lengthof a cycle in type j and a block of identical types in j + 1 tends to 0 rapidly, which in our casecorresponds to

sn(ln − 1)2jqn(ln − 1)2j+2qn

=sn

(ln − 1)2→ 0 as n→∞.

Hence, we can apply the analysis in [Fe76, Section 5] or [ORW82, Chapter 10] to show that thereal-analytic realization T ∈ Diff ω

ρ (T2, µ) of this strongly uniform circular system is ergodic andnot loosely Bernoulli.

To show that there even exists an uncountable family of diffeomorphisms in Diff ωρ (T2, µ) which

are pairwise not Kakutani-equivalent we follow the approach in [ORW82, Chapter 12]. Let a =(an)n∈N be an infinite sequence of zeros and ones. If an+1 = 0, we construct the (n+ 1)-blocks asdescribed above. In case of an+1 = 1, we define the (n+ 1)-blocks in the construction sequence ofthe odometer-based system as

w(n+1)j =

((w(n)sn

)(ln−1)2j−1 (w

(n)sn−1

)(ln−1)2j−1

. . .(w

(n)1

)(ln−1)2j−1)(ln−1)2·(sn+1−j)+1

,

9.2 Real-analytic non-Bernoulli diffeomorphisms with property K 45

i. e. we run through the types of n-blocks in a cycle in decreasing order this time. As above,we consider the corresponding strongly uniform circular system and realize it as a diffeomorphismTa ∈ Diff ω

ρ (T2, µ). As shown in [ORW82, Chapter 12, Proposition 3.4], if an 6= bn for infinitelymany n ∈ N, then Ta and Tb are not Kakutani-equivalent. Certainly, this gives an uncountablefamily and proves Theorem D.

9.2 Real-analytic non-Bernoulli diffeomorphisms with property K

We follow along the lines of [Ka80], where the first C∞ example of a non-Bernoulli diffeomorphismwith property K was constructed. For that purpose, let A be a hyperbolic automorphism of T2

and let S ∈ Diff ωρ (T2, µ) be an ergodic and not loosely Bernoulli diffeomorphism as constructed in

the previous Subsection. We consider its time one suspension Stt∈R on N = T2× [0, 1]/ ∼, where(x, 1) ∼ (S(x), 0). Then we define the real-analytic diffeomorphism

T : T2 ×N → T2 ×N, T (x, y) =(Ax, Sϕ(x)(y)

)with a real-analytic map ϕ : T2 → R which is not cohomologous to a constant (i. e. ϕ cannot bewritten as ϕ = ψ A − ψ + c with measurable ψ : T2 → R and c ∈ R). Then T has propertyK by [Ka80, Theorem 1]. To see that T is not Bernoulli we consider the special flow Aϕt t∈R onMϕ =

(x, s) ∈ T2 × R

∣∣ 0 ≤ s ≤ ϕ(x)

and the flow Tt = Aϕt × St on Mϕ × N . Since S is not

loosely Bernoulli, Tt is not loosely Bernoulli. Moreover, we see that T is the Poincare map of Tt onM × 0 ×N . Hence, T as a section of a non-loosely Bernoulli flow cannot be loosely Bernoulli.

Acknowledgement

The authors of this paper express their gratitude towards Anatole Katok for asking the questionthis paper answers. He also helped the authors with numerous mathematical discussions and asteady stream of encouragement.

Also this paper would not be possible without the discussions the authors had with MatthewForeman who patiently explained several intricacies of his delicate work and took an interest towardsthe current result. He also showed the first authors how to produce diagrams using spreadsheetsoftware. The second author also thanks Marlies Gerber for several discussions in the realm ofanti-classification results.

References 46

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