aa r X i v : . [ m a t h . D S ] J un COBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS
SHREY SANADHYA
Abstract.
Let S and T be two aperiodic commuting automorphisms of a stan-dard Borel space ( X, B ) , and Cob ( S ) , Cob ( T ) be the sets of their real valued(Borel) coboundaries. We show that Cob ( S ) = Cob ( T ) if and only if S = T ± .We also prove a weaker form of Rokhlin Lemma for Borel Z d -actions. Contents
1. Introduction 12. Preliminaries 33. Weak Rokhlin Lemma for Borel Z d -actions 54. Borel automorphisms with same coboundaries 10References 141. Introduction
Let T be an aperiodic Borel automorphism of a standard Borel space ( X, B ) . Areal valued Borel map f on X is called a (Borel) coboundary for T if there exists aBorel function g such that f ( x ) = g ( x ) − g ( T x ) (1.1)for every x ∈ X . Two Borel functions f and h are called cohomologous if f − h isa coboundary. We denote by Cob ( T ) the set of real valued coboundaries of T .The key question we ask in this note is the following: To what extent does the set Cob ( T ) determine T ? In particular we ask, when can two aperiodic Borel automor-phisms of ( X, B ) have the same set of (Borel) coboundaries? Our work is motivatedby results in the ergodic theory and Cantor dynamics. I. Kornfeld ([Kor99]) an-swered the above question for ergodic measure preserving transformations: Theorem 1.1 (Kornfeld [Kor99]) . Suppose σ and τ are two commuting invert-ible ergodic measure preserving transformations of a non-atomic probability space ( X, B , µ ) . They have the same coboundaries if and only σ = τ ± . Mathematics Subject Classification.
Key words and phrases. coboundries, Rokhlin Lemma, Borel Z d -actions. The coboundries in Theorem . are measurable real valued functions that satisfy (1 . for µ a.e. x ∈ X . A similar result in the context of Cantor dynamics is due toN. Ormes (Theorem . ). We refer the reader to [Orm00] for details. Coboundariesconsidered in Theorem . are continuous. Theorem 1.2 (Ormes [Orm00]) . Let ( X, T ) and ( Y, S ) be two Cantor minimalsystems. There is an orbit equivalence h : X → Y which induces a bijection fromthe set of real S -coboundaries to the set of real T -coboundaries if and only if S and T are flip conjugate, i.e. S is conjugate to T or S is conjugate to T − . There exist many similarities between Borel dynamics and ergodic theory butthe fact that in Borel dynamics we do not consider any prescribed measure on theunderlying space makes it significantly different from ergodic theory. For exam-ple, in the study of Borel dynamics we consider maps that are defined everywhere(instead of a.e. in ergodic theory).Our motivation to study cohomology of a Borel dynamical system stems from thefact that a similar study in ergodic theory and topological dynamics have provedto be very useful. Cohomology has been used in the study of measure theoreticdynamical systems up to orbit equivalence. It has given rise to methods whichhave been widely applicable in classification of dynamical systems. A completereference to papers focused on the study of cohomolgy in this context is too longto mention. We list few key reference which include the seminal work of Ramsay[Ram71], Moore [Moo70], Schmidt [Sch77, Sch90] and Zimmer [Zim84].In this paper we prove a result that is similar to Theorem . and Theorem . in Borel context (Theorem . ). The main result of the paper is following: Theorem 1.3.
Let
S, T be two aperiodic, commuting Borel automorpisms of stan-dard Borel space ( X, B ) . Then S and T have same set of (Borel) coboundaries ifand only if S = T ± . A key component in the proof of Theorem . is a two-dimensional version of thefollowing theorem (also see [BDK06, Lemma 3.3]): Theorem 1.4 (Weiss [Wei84]) . Let S be an aperiodic Borel automorpisms on astandard Borel space ( X, B ) . Then there exists a decreasing sequence A n , n ∈ N , ofBorel sets such that ( i ) for each n , ∞ S i = −∞ S i A n = ∞ S i = −∞ S i ( X − A n ) = X, ( ii ) for each n , the sets A n , S ( A n ) , S ( A n ) , ...S n − A n are pairwise disjoint, ( iii ) the intersection ∞ T n =1 A n := ∅ . The nested sequence of Borel sets A n , satisfying conditions ( i ) − ( iii ) of Theorem . are called a vanishing sequence of markers . We prove a finite dimensional versionof this theorem (Theorem . ).Let Γ be a countable group of Borel automorphisms of a standard Borel space ( X, B ) , then a Borel set A ⊂ X is a complete section for Γ if every Γ -orbit intersectsthe set A . The following theorem is a finite dimensional version of Theorem . . OBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS 3
Theorem 1.5.
Let { T i } , i = 1 , , .., d be aperiodic commuting Borel automorphismsof a standard Borel space ( X, B ) . Let Γ = { T i T i ...T i d d : i , i , ..., i d ∈ Z } be thecountable automorphism group generated by { T i } such that Γ acts freely. Then forany d -tuple ( t i ) of positive integers, and any ǫ > , there exists a Borel set A , suchthat ( i ) A is a complete section for Γ , ( ii ) the sets T k ...T k d d A , ≤ k i < t i , i = 1 , , .., d , are pairwise disjoint. Theorem . can also be considered as a weaker version of Rokhlin Lemma forBorel Z d -action.The outline of the paper is as follows. In Section , we provide basic definitionsand preliminary results about groups of Borel automorphisms. In Section , weprovide proof of Theorem . . In Section , we use Theorem . (for d = 2 ), toprove Theorem . .Throughout the paper, we use the following notation : • ( X, B ) is a standard Borel space with the σ -algebra of Borel sets B = B ( X ) . • A one-to-one Borel map T of space ( X, B ) onto itself is called a Borel au-tomorphism of X . In this paper the term "automorphism" means a Borelautomorphism of ( X, B ) . • Aut ( X, B ) is the group of all Borel automorphisms of X with the identitymap I ∈ Aut ( X, B ) . • A countable subgroup Γ of Aut ( X, B ) is called a group of Borel automor-phisms. 2. Preliminaries
In this section we provide the basic definitions from Borel dynamics and descrip-tive set theory.
Borel dynamical system.
Let X be a separable completely metrizable topologicalspace (also called Polish space ). Let B be the σ -algebra generated by the open setsin X . Then we call the pair ( X, B ) a standard Borel space . Any two uncountablestandard Borel spaces are isomorphic. Any countable subgroup Γ of Aut ( X, B ) iscalled a Borel automorphism group , and the pair ( X, Γ) is referred to as a Boreldynamical system.
In this paper we will only work with countable subgroups of
Aut ( X, B ) . Definition 2.1.
Let G be any countable group and e be the identity of G . Forevery g ∈ G , define a map ρ g : X → X , such that ( i ) ρ gh ( x ) = ρ g ( ρ h ( x )) for every g, h ∈ G , ( ii ) ρ e ( x ) = x for every x ∈ X and ( iii ) x ρ g ( x ) is a one-to-one, ontoBorel map for each g ∈ G . Then ρ ( G ) = { ρ g : g ∈ G } is called a Borel action ofthe group G on ( X, B ) .Note that ρ ( G ) is a countable subgroup of Aut ( X, B ) . We say that ρ is a freeaction (of group G ) if ρ g ( x ) = x for some x ∈ X , implies g = e . Thus a free action ρ results in an injective group homomorphism φ : G → Aut ( X, B ) : g ρ g ( x ) . Any SHREY SANADHYA
Borel automorphism T ∈ Aut ( X, B ) gives rise to a Borel action of group Z (alsocalled Borel Z -action) by identifying k ∈ Z with T k ∈ Aut ( X, B ) .In the study of Borel dynamical systems the theory of Countable Borel equivalencerelation (CBER) on ( X, B ) plays an important role as it provides a link betweendescriptive set theory and Borel actions (see Theorem . ). We call an equivalencerelation E on ( X, B ) Borel if it is a Borel subset of the product space E ⊂ X × X .We say that an equivalence relation E on ( X, B ) is countable if every equivalenceclass [ x ] E := { y ∈ X : ( x, y ) ∈ E } is a countable set for all x ∈ X . Let B ∈ B be aBorel set, then the saturation of B with respect to a CBER E on ( X, B ) is the setcontaining entire equivalence classes [ x ] E for every x ∈ B .Let Γ be a Borel automorphism group of ( X, B ) , then the orbit equivalence rela-tion generated by the action of Γ on X is given E X (Γ) = { ( x, y ) ∈ X × X : x = γy for some γ ∈ Γ } . Note that E X (Γ) is a CBER. We will call an equivalence relation E periodic ata point x ∈ X if the equivalence class [ x ] E is finite. Similarly, an equivalencerelation E is aperiodic at x ∈ X if the equivalence class [ x ] E is countably infinite.If every E -class is countably infinite we will say that the equivalence relation E is aperiodic . In this paper we work with aperiodic CBERs. The study of aperiodicCBERs in itself is an area of immense importance in descriptive set theory. Werefer our readers to [Kec19] for an up-to-date survey of the theory of countableBorel equivalence relations.The following theorem shows that all CBERs come from Borel actions of count-able groups. Theorem 2.2 (Feldman–Moore [FM77]) . Let E be a countable Borel equivalencerelation on a standard Borel space ( X, B ) . Then there is a countable group Γ ofBorel automorphisms of ( X, B ) such that E = E X (Γ) . A Borel set C is called a complete section for an equivalence relation E on ( X, B ) if every E -class intersects C , in other words [ C ] E = X . Let Γ ∈ Aut ( X, B ) be acountable Borel automorphism group. Then we will denote by C Γ the collection ofBorel subsets C such that C and X \ C both are complete section for E X (Γ) .A CBER, E on ( X, B ) is hyperfinite if E = S ∞ i =1 E i , where each E i is a finiteBorel equivalence relation and E i ⊂ E i +1 for all i ∈ N . If G is a countable groupsuch that E X ( G ) = E and it acts freely, then we call G to be a hyperfinite groupof Borel automorphisms of ( X, B ) .Let Γ , Γ be two countable Borel automorphism groups of ( X, B ) . We say that Γ and Γ are orbit equivalent (also o.e. ) if there exists a Borel isomorphism φ : X → X such that φ (Γ x ) = Γ ( φ ( x )) , ∀ x ∈ X . In other words Γ orbit of x is same asthe Γ orbit of φ ( x ) for every x ∈ X . The following theorem gives an importantcharacterisation of a hyperfinite CBER. Theorem 2.3 (Slaman-Steel [SS88], Weiss [Wei84]) . Suppose E is a CBER. Thefollowing are equivalent: . E is hyperfinite. OBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS 5 . E is generated by a Borel Z -action. Weak Rokhlin Lemma for Borel Z d -actions In this section we prove Theorem . which can be considered as a weak versionof Rokhlin Lemma for Borel Z d -action. Before we prove Theorem . it is useful todiscuss its one-dimensional version (Theorem . ) due to B. Weiss ([Wei84]). Wehave included the proof of Theorem . for completion. The proof of Theorem . is from [Nad13, chapter 7]. Definition 3.1.
Let S be an aperiodic Borel automorphism of a standard Borelspace ( X, B ) . A Borel set W ∈ B is said to be wandering with respect to S if thesets S i W , i ∈ Z , are pairwise disjoint. We will denote by W S (or W when S isobvious) the sigma ideal generated by all the wandering sets in B .By Poincare Recurrence Lemma, we can say that given S ∈ Aut ( X, B ) and A ∈ B , there exists N ∈ W S such that for each x ∈ A \ N , the points S n x returnto A for infinitely may positive n and also for infinitely many negative n . Theelements in A \ N are called non-wandering or recurrent elements of A . Definition 3.2.
Let A ∈ B be a Borel set such that every point of A is non-wandering with respect to S ∈ Aut ( X, B ) . The induced automorphism on A , de-noted by S A is defined by S A ( x ) = S n ( x ) , x ∈ A , where n = n ( x ) is the smallestpositive integer such that S n ( x ) ∈ A .Lemma . shows that given a Borel automorphism S of a standard Borel space ( X, B ) , we can find a family of generating sets for B such that none of the sets inthe family contains a full S -orbit. Lemma 3.3 (Nadkarni [Nad13]) . Let S be a free Borel automorphism on a standardBorel space ( X, B ) . Then there exists a countable collection of Borel sets { A } i ∈ N such that it generate B and none of the set in the collection contains a full S -orbit.Proof . Since ( X, B ) is standard Borel space, it is countably generated and countablyseparated. Let { A i } i ∈ N , be a separating system of sets which are closed undercomplement and generate the sigma algebra B . Note that for each i ∈ N , the set B i = ∩ ∞ k = −∞ S k A i is the largest subset of A i that is invariant under S . Since thecollection A i separates points, given any x ∈ X there is at least one i ∈ N such that x ∈ A i \ B i := C i (say). Otherwise, there exists some x ∈ X such that x ∈ B i whenever x ∈ A i . So, whenever x ∈ A i , orb ( x , S ) ⊆ B i ⊆ A i (here orb ( x, S ) denotes the orbit of x under S ). This is a contradiction since S is free, orb ( x , S ) has more than one point, while the collection { A i } i ∈ N separates points. Thusthe countable collection of sets C , C , C , ... together with the collection B i ∩ C j , i, j ∈ N , generates B and none of the sets in this collection contains a full S -orbit.We rename this collection { A i } i ∈ N to obtain the statement of the lemma. (cid:3) Definition 3.4.
A set A ∈ B is said to be decomposable (mod W S ) if we can write A as a disjoint union of two Borel sets C and D such that saturation (mod W S ) of C , D and A with respect to S are same. In other words, SHREY SANADHYA ∞ [ i = −∞ S i C = ∞ [ i = −∞ S i D = ∞ [ i = −∞ S i A (mod W S ) . Lemma 3.5 (Nadkarni [Nad13]) . Every set in B is decomposable (mod W S ) where S is a free Borel automorphism on a standard Borel space ( X, B ) .Proof . We first show that X is decomposable (mod W S ). Let { A i } i ∈ N be a familyof Borel sets closed under complement that generate B . In light of above discussionwe can assume that no A i contains a full S -orbit. Let B = A C = (cid:18) ∞ S i = −∞ S i A (cid:19) \ A B = A \ (cid:18) ∞ S i = −∞ S i A (cid:19) C = (cid:18) ∞ S i = −∞ S i B (cid:19) \ B . .. .. . B n = A n \ ∞ S i = −∞ S i (cid:18) n − ∪ i =1 B i (cid:19)! C n = (cid:18) ∞ S i = −∞ S i B n (cid:19) \ B n ... ...... ...Now put B = ∞ ∪ i =1 B i and C = ∞ ∪ i =1 C i . Note that B and C are disjoint and theirunion is X (this is true since we chose the collection { A i } i N to be closed undercomplement). Also note that ∞ [ i = −∞ S i B = ∞ [ i = −∞ S i C = X. Thus X is decomposable (mod W S ). To show that any Borel set A ∈ B isdecomposable (mod W S ) we apply this result with induced automorphism S A . (cid:3) Theorem 3.6 (Weiss [Wei84]) . Let S be an aperiodic, Borel automorpisms on astandard Borel space ( X, B ) . Then there exists a decreasing sequence A n , n ∈ N , ofBorel sets such that ( i ) for each n , ∞ S i = −∞ S i A n = ∞ S i = −∞ S i ( X − A n ) = X , in other words, for each n , A n ∈ C S , ( ii ) for each n , the sets A n , S ( A n ) , S ( A n ) , ...S n − A n are pairwise disjoint, ( iii ) the intersection ∞ T n =1 A n =: A ∞ is a wandering set.Proof . By the lemma above choose A such that A , ( X − A ) decompose X (in otherwords A ∈ C S ) and set A = A . For each x ∈ A , let n ( x ) be the first positiveinteger n such that S n ( X ) / ∈ A . Since A does not contain any full S -orbit, theexistence of such a positive integer n ( x ) is guaranteed for each x ∈ A . Write OBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS 7 E k = { x ∈ A : n ( x ) = k } , k = 1 , , .. and put A = ∞ [ k =1 S k − E k ⊆ A . We note that ∞ S i = −∞ S i A = X and A ∪ SA = ∅ . Now we apply the same argumentto S A and obtain a set A ⊆ A such that A ∩ S A A = ∅ and the saturation of A under S A is A , hence the saturation of A under S is entire X . Continuingsimilarly we obtain A , A , ... a decreasing sequence A n , n ∈ N , of Borel sets suchthat for each n ( i ) ∞ S i = −∞ S i A n = ∞ S i = −∞ S i ( X − A n ) = X , ( ii ) A n , S ( A n ) , S ( A n ) , ...S n − A n are pairwise disjoint.This proves part ( i ) and ( ii ) . Note that the set A ∞ = ∞ T n =1 A n has the propertythat A ∞ , SA ∞ , ..., S n − A ∞ are pairwise disjoint for each n , thus A ∞ is a wanderingset. This shows ( iii ) . (cid:3) Remark . If we replace A n with B n such that B n = ( A n \ A ∞ ) ∪ | i | >n S i A ∞ , theneach point in B n is recurrent and B n +1 ⊂ B n , for every n ∈ N . Moreover we getthat ∞ T n =1 B n = ∅ .For ease of notation, we will prove Theorem . for d = 2 (see Theorem . below). Remark . For n, m ∈ N , let I ( n,m ) denote the index set containing nm pairs ofintegers i.e. I ( n,m ) = { (0 , , (0 , , ..., ( n − , m − } . Each element of I ( n,m ) represents the powers to which S and T are raised. Thus ( i, j ) ∈ I ( n,m ) corresponds to S i T j . For the rest of the section, we work withthe lexicographic order on Z (denoted by " ≺ "). For ( i , j ) , ( i , j ) ∈ Z , we say ( i , j ) ≺ ( i , j ) if, ( i ) i < i or ( ii ) i = i and j < j . Theorem 3.9.
Let Γ be a countable Borel automorphism group generated by twocommuting aperiodic Borel automorphisms S, T of ( X, B ) , i.e., Γ = { S i T j : i, j ∈ Z } . Assume that Γ acts freely. Then for each pair ( n, m ) ∈ N there exists a Borelset A ( n,m ) , such that, ( i ) for each pair ( n, m ) , A ( n,m ) ∈ C Γ , ( ii ) for each ( n, m ) , the sets S i T j A ( n,m ) , ( i, j ) ∈ I ( n,m ) , are pairwise disjoint. SHREY SANADHYA
Proof . A Borel Z d -action defines a hyperfinite equivalence relation (see [GJ15],[Wei84]). Since Γ is a Z -action, E X (Γ) is hyperfinite, hence generated by a Borel Z -action (see Theorem . ). Thus there exists R ∈ Aut ( X, B ) , such that for every x ∈ X , Γ x = { R i x : i ∈ Z } .Using lemma . , choose A ∈ C R such that A does not contain a full R -orbit.Set A (0 , = A . Let ( n, m ) be a pair of integers and let I ( n,m ) be the correspondingindex set (see Remark . ). As mentioned in Remark . above, there are nm elements in the set I ( n,m ) , each corresponding to a power of S and T . We considerlexicographic order on the index set. Hence the smallest element of I ( n,m ) is thepair (0 , (which corresponds to S T ) the next element in the order is the pair (0 , (which corresponds to S T ) and so forth. We want to find A (0 , ⊂ A suchthat, A (0 , ∈ C R and A (0 , ∩ T ( A (0 , ) = ∅ .Define { B j } j ∈ Z \{ } a partition of A = A (0 , as follows, B j = { x ∈ A (0 , : T x = R j x } , j ∈ Z \ { } . Since A (0 , does not contain a full R -orbit, none of the B j , j ∈ Z \ { } contains full R -orbit. We start with B − . For x ∈ B − , let − n ( x ) be the first negative integer − n such that R − n ( x ) / ∈ B − . Write E − − k = { x ∈ B − : − n ( x ) = − k } , k = 1 , , .. Since R − n ( x ) ( x ) / ∈ B − , there are two possibilities:(i) R − n ( x ) ( x ) ∈ X \ A (0 , (we call such elements of B − type a elements) or(ii) R − n ( x ) ( x ) ∈ B i , for i = − (we call such elements of B − type b elements).We remove type b elements from B − . Thus, B − = B − \ { x ∈ B − : x is type b } , and A (0 , = A (0 , \ { x ∈ B − : x is type b } . Hence the set E − − k becomes E − − k = { x ∈ B − : − n ( x ) = − k ; R − n ( x ) ( x ) ∈ X \ A (0 , } , k = 1 , , .. Put A − = −∞ [ − k = − R ( − k +1) ( E − − k ) ⊆ B − (3.1)Note that A − ∩ R − ( A − ) = ∅ , and R − ( A − ) ∩ B i = ∅ for i ∈ Z \ { } , i = − .Now we repeat the same process for B . For x ∈ B , let n ( x ) be the first positiveinteger n such that R n ( x ) / ∈ B . Again there are two possibilities:(i) R n ( x ) ( x ) ∈ X \ A (0 , (again we denote such elements as type a elements of B ) or OBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS 9 (ii) R n ( x ) ( x ) ∈ B i , for i = − , (type b elements of B ).We remove type b elements from B . Thus B = B \ { x ∈ B : x is type b } , and A (0 , = A (0 , \ { x ∈ B : x is type b } . The set E k is now defined as E k = { x ∈ B : n ( x ) = k ; R n ( x ) ( x ) ∈ X \ A (0 , } , k = 1 , , .. Put A = ∞ [ k =1 R ( k − ( E k ) ⊆ B . (3.2)Note that A ∩ R ( A ) = ∅ , and R ( A ) ∩ B i = ∅ for i ∈ Z \ { } , i = 1 .We now repeat this process for B − and then B and so on. Finally, for every j ∈ Z \ { } we obtain A j ⊆ B j such that, R j ( A j ) ∩ A j = ∅ and R j ( A j ) ∩ B i = ∅ where i = j , i ∈ Z \ { } . Put A (0 , = [ i ∈ Z \{ } A i ⊆ A (0 , , then A (0 , ∈ C R and A (0 , ∩ T ( A (0 , ) = ∅ . Thus, we found a set A (0 , ∈ C R such that A (0 , ∩ T ( A (0 , ) = ∅ . Now we move on tonext power in the lexicographic order, i.e. (0 , . Instead of working with A we nowwork with A (0 , ⊂ A and repeat the above procedure to find A (0 , ⊂ A (0 , , suchthat A (0 , ∈ C R and A (0 , ∩ T ( A (0 , ) = ∅ . Thus, we have obtained set A (0 , ⊂ A such that A (0 , ∈ C R and A (0 , ∩ T ( A (0 , ) ∩ T ( A (0 , ) = ∅ . We continue thisprocess for all nm pairs in the index set I ( n,m ) (which corresponds to powers of S and T ) and obtain set A ( n − ,m − . Rename this set as A ( n,m ) to be consistent withstatement of Theorem . . The set A ( n,m ) satisfies ( i ) and ( ii ) . This completes theproof. (cid:3) With help of an example, we illustrate the proof of Theorem . . Example 3.10.
Assume ( n, m ) = (2 , . We want to describe sets A ( i,j ) ’s ( ≤ i < , ≤ j < ), that we obtain at each step. The elements of I (2 , in lexicographicorder are { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , } (they correspond to S T = e , T , T , S , ST , ST respectively). In the first step we obtain set A (0 , such that A (0 , ∩ T A (0 , = ∅ . In the second step we obtain set A (0 , such that the sets A (0 , , T A (0 , , T A (0 , are mutually disjoint.The next element in the index set is (1 , . So we have to go from T to S , hencewe would have to work with powers of R that corresponds to ST − . In other words the partition B j will be B j = { x ∈ A (0 , : ST − x = R j x } . This step will yield set A (1 , such that sets { A (1 , , ST − A (1 , , ST − A (1 , , SA (1 , } are mutually disjoint. Similarly, the next step will yield set A (1 , such that sets { A (1 , , T A (1 , , ST − A (1 , , SA (1 , , ST A (1 , } are mutually disjoint. Finally we will obtain set A (1 , such that sets { A (1 , , T A (1 , , T A (1 , , SA (1 , , ST A (1 , , ST A (1 , } are mutually disjoint. Denote A (2 , = A (1 , , thus sets S i T j A (2 , are disjoint for ≤ i < , ≤ j < as needed. Proof of Theorem . : The proof of Theorem . is identical to the proof of Theorem . . Instead of working with -dimensional index set I n,m , we will work with d -dimensional index set (with lexicographic ordering). Everything else remains thesame. (cid:3) Borel automorphisms with same coboundaries
Let S and T be two aperiodic commuting Borel automorphisms of standard Borelspace ( X, B ) . In this section we show that S and T have same coboundaries if andonly if S = T ± (Theorem . ). We denote by Γ = { S i T j : i, j ∈ Z } , the groupgenerated by S, T and assume that it acts freely. Let µ be a Γ - quasi-invariantprobability measure on ( X, B ) .As mentioned in the introduction, I. Kornfeld proved a similar result for commut-ing ergodic transformations of a non-atomic probability space (see [Kor99, Theorem1]). Although similar in nature, our work differs from [Kor99] in the following man-ner. In [Kor99] the author used a version of Z -Rokhlin Lemma (see [Kor99, Lemma1]) which he called a weak form of Rokhlin Lemma for a measure preserving Z -action (attributed to Conze [Con73]). We use Theorem . , which can be consideredas a weak form of Rokhlin Lemma for Borel Z -actions.The other principal difference is that in [Kor99], the author worked with aninvariant ergodic measure on the ambient probability space. We do not work withany prescribed measure on the standard Borel space ( X, B ) . However, in the proofof Theorem 4.2, we use the existence of Γ -quasi-invariant measure on the space ( X, B ) by way of proof by contradiction. Remark . Note that S = T − implies Cob ( T ) = Cob ( S ) . To see this, let f ∈ Cob ( T ) , thus f ( x ) = h ( x ) − h ( T x ) for some Borel function h and every x ∈ X .Then f can also be written as f ( x ) = h ( x ) − h ( Sx ) where h = − h ◦ T . Hence,Theorem . implies Theorem . . Theorem 4.2. If Cob ( S ) ⊆ Cob ( T ) then S = T n for some n ∈ Z . OBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS 11
Proof
Assume by contradiction that S is not a power of T . We will construct aBorel function f : X → R , which is a coboundary for S and not a coboundary for T . In particular, we will construct f with following properties : ( a ) There exists a constant M ∈ R , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 f ◦ S k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M, (4.1)for every x ∈ X and n ∈ N . ( b ) Let µ be a Γ -quasi-invariant probability measure on ( X, B ) . For every r ∈ N ,there exists m r ∈ N and a set A r ∈ B , µ ( A r ) > β (for some β ∈ R + ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ r (4.2)for all x ∈ A r .Observe that (4 . implies that f ∈ Cob ( S ) . To see this, set g ( x ) = sup n ≥ n − X k =0 f ◦ S k ( x ) ! . Thus f ( x ) = g ( x ) − g ( Sx ) for all x ∈ X .The following statement is a consequence of the fact that µ is Γ -quasi-invariant:For any β > and every < ǫ < β , there exists a δ > , such that ǫ + δ < β andfor every B ∈ B with µ ( B ) < δ , we have µ ( γB ) < ǫ , for every γ ∈ Γ .We claim that (4 . implies that f / ∈ Cob ( T ) . By contradiction assume that f ∈ Cob ( T ) . Thus, there exists a transfer function g ( x ) such that f ( x ) = g ( x ) − g ( T x ) .Hence, n − P k =0 f ◦ T k ( x ) = g ( x ) − g ◦ T n ( x ) . Let β be as in ( b ) and assume < ǫ < β (as above), then for any K ∈ R , with µ ( { x : | g ( x ) | ≥ K } ) < δ , we have µ (cid:16)n x : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 f ◦ T k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ K o(cid:17) < ǫ + δ < β for any n ∈ N , contradicting (4 . .Now we construct a Borel function f with properties ( a ) , ( b ) . The function f willbe constructed as the sum of infinite series f := ∞ P r =1 f r , in which every term f r isassociated with a certain Rokhlin tower ξ r of the Z -action generated by commutingBorel automorphisms S and T . Below we describe the construction of tower ξ r andthe associated function f r .The size of tower ξ r (given by ( n r m r ) ) is determined by two increasing sequence { n r } and { m r } of natural numbers. We also associate a decreasing sequence { α r } ofreal numbers with towers ξ r . The only restriction on sequence { n r } is that it is anincreasing sequence of natural numbers. On the other hand, we have the followingassumptions for sequence { m r } and { α r } : m r (cid:16) ∞ X s = r +1 α s (cid:17) ≤ , r = { , , ... } (4.3) m r α r ≥ r + 2 (cid:16) r − X t =1 α t m t (cid:17) , r = { , , ... } (4.4)... ... ... ...1.25 α r -1.25 α r α r · · · .75 α r -.75 α r .75 α r · · · α r -1.25 α r α r · · · Figure 1.
The Z tower ξ r .The sequences { m r } and { α r } with above properties can be constructed induc-tively. We use Theorem . to define tower ξ r as follows. Set n = n r and m = m r inTheorem . to obtain a set A r = A ( n,m ) , which is the base of tower ξ r (the bottomleft block in Figure 1). This tower is a rectangle made up of n r m r disjoint blockseach representing a set of the form S i T j A r , where ≤ i < n r and ≤ j < m r (seeFigure 1). The horizontal direction in the tower corresponds to the transformation S and the vertical corresponds to the transformation T .Function f r is zero outside the tower ξ r . On the tower ξ r it is defined to beconstant on each square (in other words f r is constant on each set S i T j A r ⊂ ξ r ).The value of f r on each square is defined as follows : In the bottom row the valuealternates +1 . α r , − . α r starting with plus sign on the leftmost set. In thesecond row from the bottom it alternates + . α r , − . α r starting with plus signon the leftmost set. In the third row it again alternates +1 . α r , − . α r , startingwith plus sign on the leftmost set and so forth (see Figure 1).We now estimate the sums (cid:12)(cid:12)(cid:12)(cid:12) n − P k =0 f r ◦ S k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12) n − P k =0 f r ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) for fixed r . Thefirst sum will be estimated from below to prove that f satisfies property ( a ) andthe second sum will be estimated from above to show that f satisfies property ( b ) .Note for any x ∈ ξ r , (cid:12)(cid:12)(cid:12)(cid:12) n − P k =0 f r ◦ S k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ α r . Thus OBOUNDARIES OF COMMUTING BOREL AUTOMORPHISMS 13 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 f ◦ S k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 ∞ X r =1 f r ! ◦ S k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X r =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k =0 f r ◦ S k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X r =1 α r =: M Therefore f satisfies property ( a ) .To see that f satisfies ( b ) , note that to every r ∈ N we have associated a pair ofnatural number ( n r , m r ) and a tower ξ r with base A r . Since A r is a complete sectionwith respect to Γ and µ is a Γ -quasi-invariant probability measure, µ ( A r ) > . Wework with A r and the corresponding tower ξ r . Note that (cid:12)(cid:12)(cid:12)(cid:12) m r − P k =0 f r ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ m r α r ,for all x ∈ A r . Thus for all x ∈ A r , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f r ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − r − X t =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f t ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∞ X s = r +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f s ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.5)Since f t is zero outside the tower ξ t and t < r , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f t ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m t − X k =0 f t ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < α t m t . Hence, − r − X t =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f t ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > − r − X t =1 α t m t . Similarly, − ∞ X s = r +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f s ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > − ∞ X s = r +1 α s m r . Thus by (4 . , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ m r α r − r − X t =1 α t m t − ∞ X s = r +1 α s m r . (4.6)Hence by (4 . and (4 . , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m r − X k =0 f ◦ T k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ r − ∞ X s = r +1 α s m r ≥ r + 32 ≥ r. (4.7)This shows that f satisfies property ( b ) , which completes the proof. (cid:3) Acknowledgments.
I would like to thank my co-advisor, Sergii Bezuglyi forsuggesting this problem and many helpful discussions. I am thankful to Paul Muhly,Palle Jorgensen and Charles Frohman for their encouragement and interest in this work. I was supported in parts by University of Iowa, Graduate College Post-Comprehensive Research Fellowship.
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Department of Mathematics, University of Iowa, Iowa City, 52242 IA, USA
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