ANY TWO IRREDUCIBLE MARKOV CHAINS ARE FINITARILY ORBIT EQUIVALENT

Transcription

1 ANY TWO IRREDUCIBLE MARKOV CHAINS ARE FINITARILY ORBIT EQUIVALENT MRINAL KANTI ROYCHOWDHURY DANIEL J RUDOLPH DEPARTMENT OF MATHEMATICS COLORADO STATE UNIVERSITY CO 8053, USA Abstract Two invertible dynamical systems (X, A, µ, T ) and (Y, B, ν, S) where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping φ from a subset X 0 of X of full measure to a subset Y 0 of Y of full measure such that φ X0 is continuous in the relative topology on X 0, φ Y0 is continuous in the relative topology on Y 0 and φ(orb T (x)) = Orb S φ(x) for µ-ae x X In this article a finitary orbit equivalence mapping is shown to exist between any two irreducible Markov chains Introduction The theory of orbit equivalence originated in the late 950 s H Dye pioneered the early development of the theory in his article [D] On groups of measure preserving transformations I In this work he proved one of the most fundamental theorems in ergodic theory, that any two invertible ergodic measure preserving transformations acting on non-atomic Lebesgue probability spaces are orbit equivalent, ie there is a measure isomorphism between the spaces taking complete orbits to complete orbits, modulo the underlying probability measures This result is usually [HO] proved by taking the binary odometer, whose ergodic structure is well understood, as one of the transformations There has been considerable interest in orbit equivalence theory since Dye s original article, and many other aspects of orbit equivalence have been studied In particular the broad study of finitary orbit equivalence was begun by Hamachi and Keane [HK] Two invertible dynamical systems (X, A, µ, T ) and (Y, B, ν, S) where X, Y are metrizable spaces, T, S are homeomorphisms on X and Y, and µ and ν are invariant Borel probability measures, are said to DJ Rudolph supported in part by NSF grant DMS

2 M K ROYCHOWDHURY AND D J RUDOLPH be finitarily orbit equivalent if there exists an invertible measure preserving mapping φ from a subset X 0 of X of measure one to a subset Y 0 of Y such that φ X0 is continuous in the relative topology on X 0, φ Y0 is continuous in the relative topology on Y 0 and φ(orb T (x)) = Orb S φ(x) for µ-ae x X It remains an open problem as to whether any two orbit equivalent systems are finitarily orbit equivalent In 003, in the article [HK] Hamachi and Keane constructed the first finitary orbit equivalence mapping between the binary odometer and the ternary odometer In this article we prove: Main Theorem: Any two irreducible Markov chains are finitarily orbit equivalent The finitary theory began in the study of isomorphism and this background can be found in [KS], [KS] Background for the methods used in the finitary orbit equivalence theory are in [HK], [HKR] Particularly note the result in [R] that any two odometers are finitarily orbit equivalent and [R] that an irrational rotation of the circle is finitarily orbit equivalent to the binary odometer This work is closest to what we do here The path to our main theorem is as follows Suppose there are two given irreducible Markov chains M, M with entropies h, h We will construct two irreducible Markov chains M n,q,α and M n,q,α with these same entropies h and h Since any two irreducible Markov chains of equal entropy are finitarily Kakutani equivalent [RR], M is finitarily orbit equivalent to M n,q,α and M is finitarily orbit equivalent to M n,q,α Since finitary orbit equivalence is an equivalence relation, it is enough to show that M n,q,α and M n,q,α are finitarily orbit equivalent For these explicit Markov chains we will construct the finitary orbit equivalence directly The paper is arranged as follows In Section we construct the Markov chains M n,q,α and M n,q,α In section 3 we define the notions of cylinders, tower partitions and establish basic results for them Section 4 discusses tower maps and Section 5 completes the finitary construction

3 FINITARY ORBIT EQUIVALENCE 3 Construction of the Markov chains M n,q,α and M n,q,α Let us first consider a Markov chain M n,q,α with state space Σ = {, a, a,, a n } and transitional probabilities A(, ) = A(, a i ) = n A(a i, ) = q { α( q) if i = j A(a i, a j ) = ( α)( q) n if i j where i, j =,,, n a a a n Figure We have omitted the probabilities of the transitions and some arrows in figure to keep the figure simple What one has here is a collection of states a i in which all pairs communicate with a leakage out to the special state with probability q, but among themselves the transitions are iid weighted so that, given one stays in this collection, the conditional probability of a state a i returning to itself is α and the transitions to the other states are all the same The special state returns to itself with probability / and jumps to each of the other states with the same probability

4 4 M K ROYCHOWDHURY AND D J RUDOLPH Here 0 < q, 0 α and n Let ρ(s) represent the initial probability of a state s Σ From symmetry it is clear that ρ(a ) = ρ(a ) = = ρ(a n ) Since the states a, a,, a n are symmetric collapsing them in one state, say Ā, we get the following Markov chain: q Ā q Figure The state space is {, Ā} with transition probabilities A(, ) = A(, Ā) = A(Ā, Ā) = q and A(Ā, ) = q Let ρ() = x, ρ(ā) = y and we have [ x y ] [ q ] = [ x y ] q

5 FINITARY ORBIT EQUIVALENCE 5 This gives x = qy Since x + y =, we have x = Hence in our Markov chain M n,q,α we have ρ() = q + q and ρ(a ) = ρ(a ) = = ρ(a n ) = q +q and y = +q n( + q) Therefore we can calculate the entropy, h(m n,q,α ) = q [ + q log n + ] log + q [α( q) log α( q) + q log q] Now and + ( α)( q) + q h(m n,q, ) = q + q [ log ( α)( q) log n n + log ] + q [( q) log ( q) + q log q], h(m n,q, ) = q n lim q 0 h(m n,q, ) = 0 + q + q [ log n + log [ q n log q n ] ] + q log q + ( n ) q + q log q n, [ lim h(m n,q, ) = q 0 n n log n + ( ] n ) log = log n n Given two Markov chains M and M with entropies h and h, without any loss of generality let us assume that 0 < h h In our Markov chain M n,q,α choose n st 0 < h h < log n Since if q 0, then h(m n,q, ) 0 and h(m n,q, ) log n n, choose q st h(m n,q, ) < h h < h(m n,q, ) n Since h(m n,q,α ) is continuous in α, choose α and α between and n st h(m n,q,α ) = h and h(m n,q,α ) = h

6 6 M K ROYCHOWDHURY AND D J RUDOLPH Let σ, σ represent the shift automorphisms and µ, µ represent the Markov measures of the Markov chains M n,q,α and M n,q,α respectively This completes the construction of the two explicit Markov chains M n,q,α and M n,q,α What makes these two chains special for our purposes is that if we collapse all the states a,, a n to the single state A then they become identical Hence we can use the original ideas of Keane and Smorodinsky to construct identical skeleta in the two Markov chains 3 Cylinders, tower partitions, basic lemmas and proposition In this section we give the definitions of canonical cylinder partition, tower partition, basic lemmas and propositions for the Markov chain M n,q,α Although in this discussion we keep n and q fixed and allow α to be either α or α, we will assume µ be to be the Markov measure and σ to be the shift automorphism on M n,q,α Let n= Σn represents the set of all possible words in the Markov chain M n,q,α For each k =,, 3, a k-skeleton S is a string of s and spaces (which represent occurrences of an unspecified a i ) written here as a word in two symbols {, } st k S = k 0 k k m km l l l m Where (i) l t ( t m) and only a i s and never an occur in the spaces, (ii) k i < k min{k 0, k m } ( i m ) Here l + (k 0 + k m ) where l = l + k + l + + k m + l m is called the length of the k-skeleton The k-skeleton of a sequence x in a Markov chain M n,q,α is found by beginning at index 0 in the sequence and reading to both the left and right until on each side a string of at least k consecutive s is found that together are not part of a single consecutive string We seek a pair of such strings where one begins at an index i 0 and the other begins at an index j > 0; all s in the strings are included in the k-skeleton of x Almost every point in these Markov chains contains infinitely many finite strings of s of length at least k in both its forward and backward orbit, so such a k-skeleton can be found with probability

7 FINITARY ORBIT EQUIVALENCE 7 Notice that a (k + )-skeleton could already be a k-skeleton if between the two strings of s one never saw an isolated string of k s If the spaces of a k-skeleton are filled in by the symbols from Σ {} in some allowable way, the resulting block having the same length as the skeleton is called a filled k-skeleton associated to the k-skeleton Clearly for a fixed k-skeleton the corresponding filled k-skeletons are either disjoint or identical We say that a k-skeleton appears in a point x, if there is a filled k-skeleton that contains the point x The filled k-skeleton that contains the point x is called the filled k-skeleton of x and is denoted by F k (x) We say that S is a subskeleton of the skeleton S, exhibited above if S is a k -skeleton for some k < k with the following configuration: S = kt l t+ k t+ for suitable t < t m l t+ k t l t Definition 3 η-canonical cylinder (η a filled k-skeleton) Let η = F k (x) be the filled k-skeleton of x This is a finite word of length l + (k 0 + k m ) The 0th coordinate of the infinite word x is at some position 0 i < l + k 0 in F k (x) The set h k (η, i) = { x F k (x) = η & x j = η i+j, 0 i + j < l + (k 0 + k m ) } is called an η-canonical cylinder Note that h k (η, 0) σ h k (η, ) σ σ h k (η, l + k 0 ) By a k-canonical cylinder (k ) we mean an η-canonical cylinder where η is a filled k-skeleton and by a canonical cylinder or just a cylinder we mean an η-canonical cylinder for some filled k-skeleton η for some k Definition 3 Canonical cylinder partition Fix a fixed k the collection C of all k-canonical cylinders, denoted by H k, is called the k-canonical cylinder partition If k is not fixed we just call it a canonical cylinder partition Notation 33 Let S k = { x M n,q,α x i =, i = 0,,,, k } For any x M n,q,α, let l k (x) be the largest value 0 with σ l k(x) x S k (or l k (x) = ) and r k (x) be the smallest value > 0 with σ r k(x) x S k (or r k (x) = ) Lemma 34 For ae x, l k (x) > and r k (x) <, but lim k l k (x) = and lim k r k (x) = Furthermore max({µ(c) c is a k-canonical cylinder}) 0 as k

8 8 M K ROYCHOWDHURY AND D J RUDOLPH Lemma 35 Every k-canonical cylinder is a disjoint union of (k + )- canonical cylinders Proof Take c to be a k-canonical cylinder defined on indices from j to j + K Now for ae x c we will have l k+ (x) < j, r k+ (x) > j+k as the cylinder c is a k-canonical cylinder Now set c x to be the (k + )-canonical cylinder containing x Hence c x c and c = x c c x almost surely As for x, y c either c x = c y or c x c y =, c is almost surely a disjoint union of (k + )-canonical cylinders Definition 36 Tower partition A tower partition is a canonical cylinder partition C endowed with an equivalence relation in which each equivalence class consists of η- canonical cylinders for some fixed filled k-skeleton η, where η and k may vary from class to class For any two η-canonical cylinders c = h k (η, i), c = h k (η, j) there is a unique t(c, c ) = t = j i Z st σ t (c ) = c We endow each equivalence class of a tower partition with the order inherited from the value t ie c < c if and only if σ t (c ) = c and t > 0 Although we refer to this as an order on the equivalence class it is important that we keep in mind not only the order but the actual translation value t(c, c ) Each equivalence class C of the tower partition C is called a tower or a column of C The number of elements in C is called the height of the tower C By the support of C we mean Supp(C) = c c C We extend this idea to any collection of sets, indicating the union of its elements as its support Lemma 37 Suppose c, c are η-canonical cylinders where η is a filled k-skeleton Suppose further that c c is an η -canonical cylinder, η a filled k -skeleton, k > k There is then a unique η -canonical cylinder c c with t(c, c ) = t(c, c ) Proof Set c = σ t(c,c ) (c ) As c is an η -canonical cylinder and the domain of indices defining c is a superset of those defining c, the translation t(c, c ) cannot place the 0th coordinate of c outside this defining domain of indices, and hence c is an also η -canonical cylinder Moreover the minimal translation carrying c to c must be t(c, c ) since if some smaller translation accomplished this it would carry c to c as well

9 FINITARY ORBIT EQUIVALENCE 9 Definition 38 k-canonical tower partition P k (k ) The canonical cylinder partition H k denoted by = st h k (η, i) = h k (η, i ) iff η = η, with an equivalence relation is called the k-canonical tower partition, denoted by P k, and is endowed with increasing order up the towers as induced by the translation t(h k (η, i), h k (η, i )) = i i Lemma 39 Suppose P k is a refining sequence of countable partitions of a probability space (X, A, µ) Suppose lim Sup k,c Pk µ(c) = 0 Then for any probability vector ρ = {ρ, ρ, ρ 3, } there is a partition Q = {q, q, q 3, } of X where each q i is almost surely a disjoint union of sets from k= P k and µ(q i ) = ρ i Proof We will inductively define the collection of sets Si k, Si k S k+ i where each Si k is a collection of elements of P, P,, P k, all the sets in these collections are disjoint and ρ i ρ k i, where ρ k i = c S k i µ(c) To start the induction we include P 0, the trivial partition and set all Si 0 = φ Suppose the collections Si k are defined Let c i, c i, P k+ be a listing of all elements of P k+, not contained in i c S k i c List them in order of decreasing measure First look for a value i with ρ i ρ k i µ(c i ) If it exists, take the least such i and add c i to Si k to start the construction of S k+ i This increases ρ k i to ρ k i + µ(c i ) ρ i If no such i exists then c i will not be assigned Proceed to c i, assigning it to the first Si k into which it will fit under the ρ i bound on measure Continue through all the c ij This completes step k+ of the induction We continue the induction through all the levels k and set S i = k= Sk i The S i s will consist of disjoint sets as desired It remains to show that for q i = c Si c that ρ i = µ(q i ) = c S i µ(c) We certainly have ρ i c S i µ(c) Suppose for some i 0, ρ i0 > c S i0 µ(c) and assume i 0 is the least such value Let a = ρ i0 c S i0 µ(c) > 0 As lim (Sup c P k k µ(c)) = 0 for k large enough and all c P k, µ(c) a But this means at such a level k of the induction any c ij can be added to S k i 0, as c S µ(c) k i 0 c S i0 µ(c) = ρ i0 a Nonetheless after this step there is still a deficit of

10 0 M K ROYCHOWDHURY AND D J RUDOLPH a in Si k 0 This can only happen if no sets at level k are left unassigned, ie the collections Si k form a disjoint cover of X in which case µ(c) =, i c S k i which forces c S µ(c) = ρ i k i for all i, including i 0, which is a conflict Hence is the lemma Corollary 39 If c is a k-canonical cylinder in the Markov chain M n,q,α, and ρ = {ρ i } is a finite or countable probability vector, then c can be partitioned almost surely into subsets c i each of which is a union of canonical cylinders and for which µ(c i ) = ρ i µ(c) Definition 30 Suppose c and c are two η-canonical cylinders where η is a filled k-skeleton We know then that c = h k (η, i ) and c = h k (η, i ) and σ i i (c ) = c If A c then we refer to σ i i (A) c as the corresponding subset of c Similarly if P is a partition of c then σ i i (P ) is the corresponding partition of c Lemma 3 Suppose S, S,, S m are collections of canonical cylinders, Supp(S i ) = S i and all S i have the same measure and are disjoint For each k and filled k-skeleton η there are sets c i,η of η-canonical cylinders so that for each i m, all Supp(c i,η ) are disjoint, S i = η Supp(c i,η ) as, each c c i,η is contained in some s S i, and for each η and all i, j m, #(c i,η ) = #(c j,η ) Proof Place an order on the canonical skeleta by saying η < η whenever η is a filled k skeleton and η is a filled k skeleton and k < k and for each fixed k order the countable list of filled k-skeleta in some way as N Now suppose for all η < η we have constructed sets c i,η We describe how to construct c i,η Remove from each S i the supports of all c i,η with η < η Call the remaining set S i For each value i consider the collection d i,η of all canonical η -cylinders contained in S i which are also subsets of some s S i If n = n(i, η ) is the minimal cardinality of these sets d i,η, select n distinct η -canonical cylinders from each d i,η and call this collection c i,η This is a greedy algorithm, taking as many of these η -cylinders as we possibly can given the requirement that the #(c i,η ) should not depend on i We continue by removing the supports of the c i,η from each S i and move to the next choice for η

11 FINITARY ORBIT EQUIVALENCE Notice that if η is a filled k-skeleton and k > k then each c i,η refines as a union of k skeleta each of whose sets is contained in some s S i This procedure will guarantee that the c i,η for each η all have the same cardinality, that each set in a c i,η is a subset of some s S i and that for fixed i they have disjoint supports What remains is to see that the supports of the c i,η for fixed i cover each S i as Suppose instead of starting the induction with the first filled - skeleton we started with the first filled k 0 -skeleton for some k 0 > As we proceed the sets c i,η chosen might be different but the measure of that part of each S i covered at each step beyond this first k 0 skeleton by the union of the supports of all c i,η so far constructed would not To see this recall that each canonical j-cylinder is as a union of canonical k-cylinders for k > j Thus we can regard that part of each S i which has been covered by c i,η, η a filled j-skeleton as covered by a union of canonical k-cylinders For each filled k-skeleton η the number of canonical η-cylinders assigned to each S i is simply the maximum possible given that the same number is assigned to each and this number will be the same whether we started with -skeleta or k 0 -skeleta For each filled k-skeleton η let f i (η) be the fraction of the canonical η-cylinders contained in S i It follows from the fact that S i is a union of canonical cylinders and the ergodic theorem that for all ε > 0, once k is large enough, for all but a set of filled k-skeleta of measure < ε, the value f i (η) is within ε of µ(s i ) That is to say, the values f i (η) are all very close together In particular this implies that the union of the subsets of each S i not covered once we have exhausted the k-skeleta in the algorithm has measure at most ε It follows that the algorithm leads to a cover of almost all of all the S i Definition 3 Common extension Let C = {c 0, σ n c 0, σ n c 0,, σ n k c0 } be an ordered column of a tower partition C and c 0 c 0 be an η-canonical cylinder Then we call the ordered set {c 0, σ n c 0, σ n c 0,, σ n k c 0 } a common extension of C by c 0 A tower partition Ĉ is said to be a tower extension of a tower partition C if each column of Ĉ is a common extension of a column of C Lemma 33 Suppose C is a tower partition and S, S,, S m are collections of subsets of C for which all Supp(S i ) have the same measure and are disjoint Moreover suppose no two sets in m i= S i are in the same column There is then a tower extension Ĉ of C so that for each

12 M K ROYCHOWDHURY AND D J RUDOLPH filled k-skeleton η the number of η canonical cylinders from Ĉ contained in each Supp(S i ) does not depend on i Proof Applying Lemma 3 we can refine each collection S i, as a partition of its support, by the canonical cylinders in the sets c i,η as η varies over all filled skeleta We will have that for each η the cardinality of c i,η does not depend on i For each i and each set s S i let this refinement be p(s) We now refine all sets in the column containing s by the natural translation of p(s) and take the corresponding natural equivalence relation generated We leave all columns that do not intersect any S i as they are This creates the tower extension Ĉ we seek Definition 34 Tower refinement Let C be a tower partition Then a tower partition C is a refinement of C if each column of C is a union of columns of some tower extension Ĉ of C The order on C extends the order on Ĉ which, as we have seen corresponds to that on C Note: If C is a tower refinement of C, then each c C is contained in a unique c C, yielding a map π : C C which preserves the measure µ Whenever one partition refines another we will let π represent the inclusion map of a set in the finer partition into a set in the coarser Note also that the set Ĉ is identical with the set C, but that its equivalence relation is finer (equivalence classes of the former are subsets of the equivalence classes of the latter) To say two sets c, c C are equivalent we write c C c Lemma 35 The (k + )-canonical tower partition P k+ refines the k-canonical tower partition P k Proof By Lemma 35, H k+ refines H k and so P k+ refines P k Lemma 36 Suppose C is a tower partition and S, S,, S m are collections of subsets of C for which all Supp(S i ) have the same measure and are disjoint Moreover suppose no two sets in m i= S i are in the same column of C There is then a tower refinement C of C so that the support of the collection of equivalence classes that cover any one of the Supp(S i ) must cover all of them Proof Applying Lemma 33 we obtain a tower extension Ĉ with the property that the number of η canonical cylinders from Ĉ contained in each Supp(S i ) does not depend on i As usual we call this set of cylinders c i,η For each filled skeleton η let f i,η be some bijection of

13 FINITARY ORBIT EQUIVALENCE 3 c,η to c i,η with f,η the identity map Extend the equivalence relation Ĉ by requiring that e c,η also be related to f i,η (e) for all i Now complete the relation under transitivity and reflexivity All equivalent sets are canonical η-cylinders for the same η and so this creates a tower partition which we call C All columns of C are unions of columns of Ĉ so this is a tower refinement of C For any choice of i, j m any cylinder e c j,η is in the same C column as f i,η f j,η (e) and so the equivalence classes covering Supp(S i ) must also cover Supp(S j ) Definition 37 ( ɛ) P k -invariance Let 0 < ɛ < and k Then a tower partition C is called ( ε) P k -invariant if there exists a tower extension P k of P k such that µ(c) ε, D c D where the sum D is taken over all columns D in P k such that all cylinders c D are C-equivalent The set of all such columns D in P k in the above inequality is called the major portion of C and is denoted by Major(C) Notice that if C is ( ε) P k -invariant and k < k then C is also ( ε) P k -invariant as P k is a finer equivalence relation than P k Proposition 38 Let C be a tower partition, let 0 < ε < and k N There exists a ( ε) P k -invariant tower refinement C of C Proof Fix ε and k N For each δ > 0 let D δ be the collection of all k-canonical cylinders c with µ(c) > δ The cylinders in D δ consist of full columns of the canonical tower partition P k and each such full column can contain at most /δ sets Choose δ 0 > 0 so small that µ(supp(d δ0 )) > ε/ Cylinders in each column C i of C are η i -canonical cylinders for some filled j i -skeleton η i Choose a finite value n 0 with n 0 i= µ(supp(c i )) δ 0ε and suppose k max i n0 j i and k k Take all the columns C i (not just those with i n 0 ) with j i k, and subdivide the column into η - canonical cylinders, η a filled k -skeleton Among all these C i, combine all the k -canonical cylinders that arise from a common filled k -skeleton η For those cylinders with j i > k leave the column C i unchanged

14 4 M K ROYCHOWDHURY AND D J RUDOLPH This produces a tower refinement C of C with the property that the columns of C that come from filled k -skeleta have supports with total measure > δ 0ε This is the tower refinement C we take To show that C is ( ε) P k -invariant, consider the tower extension of P k obtained by refining its columns into filled k -skeleta This is the P k we need Call its columns D i Consider now those columns D i in P k which are not all elements of a single column of C We need to show this collection of columns has measure less than ε To do this, first exclude the columns that are not in Supp(D δ0 ), a set of measure < ε/ The remaining columns D i each have cardinality < /δ 0 If such a column is not completely contained in some column of C then at least one of its subsets is refined when we formed C and hence lies outside n 0 i= Supp(C i) The collection of all such sets in P k, refined by C has measure < εδ 0 Hence the union of all sets in all columns which contain such a refined set and also lie in D δ0 has measure < ε/ All remaining columns D i are elements of a single column of C The union of the sets we have excluded has total measure < ε and this concludes the proof that C is ( ε) P k -invariant 4 Tower maps The two pieces of the Hamachi-Keane approach to finitary orbit equivalence are the constructions of ( ε)-invariant extensions, which we have described in Proposition 38 and the construction of commuting diagrams of tower map, which we now proceed to do Definition 4 Let C and D be tower partitions of M n,q,α and M n,q,α A tower map is a surjective mapping φ : C D taking each C-equivalence class bijectively to some D-equivalence class and taking µ to µ, in that µ (d) = µ (c) (d D) { c C : φ(c)=d } Definition 4 Suppose π : C D is a tower map, as is ψ : D C where C refines C and D refines D In this case recall that there will be intermediary towers Ĉ and ˆD which extend C and D and whose equivalence classes are finer than C and D We say the diagram π C C φ D π ψ D

15 FINITARY ORBIT EQUIVALENCE 5 commutes strongly if in fact the extended diagram commutes π π C Ĉ C φ D ψ π ˆD π ψ D Proposition 43 Let φ : C D be a tower map, and let C be a tower refinement of C Then there exists a tower refinement D of D and a tower map ψ : D C such that commutes strongly π C C φ D π ψ D Proof As C is a tower refinement of C and Ĉ is a tower extension of C, each column of C is a union of columns of Ĉ Let us consider a D-equivalence class D and assume that each equivalence class in D consists of η-canonical cylinders for some fixed filled k-skeleton η, η and k may vary from class to class Let d D, d be its minimal element according to the ordering We can write () µ (c ) = µ (d) {c C : φ(π(c ))=d} Thus ρ d = {µ (c )/µ (d) : φ(π(c )) = d} is a finite or countable probability vector By Corollary 39 for each c and k > k there exists a set of filled k -skeleta η and disjoint η-canonical sub cylinders of d, which we call {I η (c, d)} st µ (c ) = µ (I η (c, d)) η Note: If for some η there are no η-canonical cylinders used then I η (c, d) = Let I(c, d) = η I η(c, d) Clearly the sets I(c, d) cover d as For all other d D we set I η (c, d ) to be the canonical translation of the sets in I η (c, d) to d All the sets in all the various I(c, d ) form a

16 6 M K ROYCHOWDHURY AND D J RUDOLPH cylinder partition For any two equivalent sets d, d D we extend the equivalence relation and order to canonically related subsets of the I η (c, d) and I η (c, d ) creating a tower extension D of D For any set d D we have d I η (c, d ) for some unique choices of η, c and d We define ψ 0 (d ) = c It is easily checked that this is a tower map and the following diagram commutes: π C φ ψ 0 Ĉ D We now use Lemma 36 to further extend D and expand the equivalence relation to create D Consider an equivalence class C in C It is a finite union of equivalence classes of Ĉ From each of these finer classes select the least element and call this list of sets e, e,, e m Set S i = ψ0 (e i ), a finite list of disjoint sets each of which is collection of canonical cylinders and all of whose supports have the same measure Moreover as ψ 0 is a tower map no two sets in the union over i of the S i are equivalent By Lemma 36 we can first extend D and then construct a tower refinement so that the equivalence classes that intersect the support of any one of the S i must contain the supports of all of them We do this successively for all the columns C of C producing the tower refinement D of D For any set d D, if d d D then we set ψ(d ) = ψ 0 (d ) It is not difficult now to see that this produces the strongly commuting diagram we seek D π 5 The finitary construction The construction of the finitary orbit equivalence between our two Markov chains M n,q,α and M n,q,α is obtained by successively applying Propositions 38 and 43 Two points are in the same column, by this we mean the canonical cylinders containing these points are in the same column Definition 5 Suppose C n is a sequence of tower partitions of M n,q,α We say this sequence generates if for all canonical tower partitions P k, once n is large enough, C n is a tower refinement of P k Proposition 5 Suppose C n (n 0) is a sequence of tower partitions of M n,q,α that generates and C n+ a tower refinement of C n There is then a set of measure one in M n,q,α such that if x and x are in this set, then they are on the same orbit if and only if there exists an n 0 such that for all n n 0, (i) x and x are in the same column of C n and (ii) if

17 FINITARY ORBIT EQUIVALENCE 7 x c n+ c n, x c n+ c n, where c n, c n C n and c n+, c n+ C n+, then t(c n, c n ) = t(c n+, c n+ ) Proof Let A be the set of all points x which for all k large enough belongs to a k-canonical cylinder and the indices in the orbit of x in these k-canonical cylinders expand in k to all of Z Such a set has measure one Let x and x be two points in A As the partitions C n refine we know that n c n = {x} and n c n = { x} If x and x are on the same orbit then for all k large enough, they are in η-canonical cylinders for the same k-canonical η and moreover the natural translation between these two cylinders does not change with k This completes one direction of the argument On the other hand, if we have such an n 0 so that for n n 0 we have t(c n, c n ) = t(c n+, c n+ ), then taking t = t(c n0, c n0 +) we have, { x} = n n0 c n = n n0 σ t c n = σ t ( n n0 c n ) = {σ t (x)} = x = σ t (x) Proof of the Main Theorem: At this point we have all the ingredients to follow the format laid down by Hamachi and Keane to construct a finitary orbit equivalence between M n,q,α and M n,q,α For completeness we include the details Constructing the finitary orbit equivalence: Choose a sequence {ε n } with 0 < ε n < and ε n < n= Using Propositions 38 and 43 inductively, produce the diagram π π π C 0 φ 0 D 0 C ψ π D C φ π D in which C 0 = {M n,q,α, } and D 0 = {M n,q,α, } are the trivial partitions and both sequences C n and D n generate We define the map φ 0 to be the tower map for which φ 0 (M n,q,α ) = M n,q,α The tower π

18 8 M K ROYCHOWDHURY AND D J RUDOLPH partition C n+ is ( ε n+ ) P mn+ -invariant and the tower partition D n is ( ε n ) P mn -invariant, where the sequence {m n } is increasing (n 0) and each square block in the diagram commutes strongly It remains to see that the maps φ n and ψ n converge to a finitary orbit map from M n,q,α to M n,q,α Let M M n,q,α consist of those points which belong to all tower partitions C n and for which both l k (x) and r k (x) are finite for all k but tend to and respectively Let M M n,q,α be the corresponding set in the second process These are both sets of full measure Let c n (x ) be the set in C n containing x M and d n (x ) be the set in D n containing x M Thus π(c n (x )) = c n (x ) and π(d n (x )) = d n (x ) As the C n and D n generate, n c n (x ) = {x } and n d n (x ) = {x } For x M we have that ψ n c n (x ) is a nested decreasing sequence of sets with ψ n+ (c n+ (x )) φ n (c n (x )) ψ n (c n (x )) for all n Thus n ψ n (c n (x )) = n φ n (c n (x )) and must consist of a single point which we call φ(x ) Symmetrically one constructs ψ : M M n,q,α and it easily follows that ψ = φ ae It follows easily that for all n and all c C n and d D n that φ(c) = φ n (c n ) as and ψ(d) = ψ n (d) as That is to say, the two maps φ and ψ are point maps that agree with all the φ n and ψ n as set maps on the subsets of full measure where they are defined As φ n and ψ n are tower maps and hence preserve measure, φ and ψ are measure preserving on all canonical cylinders and this implies they are measure preserving maps For any canonical cylinder d in M, once n is large enough we will have d is a union of sets in D n as this sequence generates This implies that φ (d) = (φ n ) (d) on the range of φ The latter is an open set and this implies φ is continuous where it is defined and symmetrically so is ψ and hence both φ and its inverse are continuous ae We now show that both φ and ψ are as bijections on orbits To begin, as n ε n < we know that for almost every x, once n is large enough, x is in the major portion of C n as a ( ε n ) P mn -invariant tower partition, and similarly for almost every x, once n is large enough, is in the major portion of D n as a ( ε n ) P mn -invariant tower partition Within these subsets of full measure in M and M we can construct further subsets M and M, which are still of full measure and for which we have the further properties that σ(m ) = M, σ(m ) = M, φ(m ) = M and ψ(m ) = M Suppose x and x = σ(x t ) are on the same orbit and both are in M Let e k (x ) and e k (x ) be the elements of P k containing x and

19 FINITARY ORBIT EQUIVALENCE 9 x respectively By Proposition 5, there is then a value k 0 and for all k k 0, the canonical translattion t(e k (x ), e k (x )) must be t Once k k 0 is large enough, x will be in the major portion of C k and this tells us that for some tower extension P k of P k the entire P k -equivalence class of e k (x ) lies in a single C k equivalence class In particular for all k large enough, x and x lie in the same column of C k Consider now the pair of points x = φ(x ) and x = φ(x ) Consider now n large enough so that c n (x ) and c n (x ) are in the same C n -column and the canonical translation between them is t This implies d n (x ) and d n (x ) are also in the same D n -column since, depending on whether n is even or odd, either c n (x ) = ψ n (d n (x )) or d n (x ) = φ n (c n (x )) and the maps ψ n and φ n are tower maps and hence bijections on equivalence classes Let the canonical translation between d n (x ) and d n (x ) be s n As the next block in the diagram strongly commutes, it commutes at the level of Ĉn and ˆD n, the tower extensions of C n and D n This implies that d n+ (x ) and d n+ (x ) must be in the same column of ˆD n and this implies that the canonical translation s n+ between d n+ (x ) and d n+ (x ) must still be s n That is to say, the canonical translations between the sets d n (x ) and d n (x ) do not change once n is large enough If we call this asymptotic value s we conclude by Proposition 5, x = σ(x s ), and that these image points are on the same orbit This implies that the φ-image of an orbit is contained in an orbit The symmetric argument implies that the ψ-image of an orbit is contained in an orbit and as ψ is the inverse of φ, we conclude φ, restricted to M is a bijection on orbits References [HO] [D] H Dye, On groups of measure preserving transformations I, Amer J Math, 8 (959), 9-59 [HK] T Hamachi and M Keane, Finitary orbit equivalence of odometers, Bull London Math Soc 38 (006) [HKR] T Hamachi, MSKeane and MK Roychowdhury, Finitary orbit equivalence and measured Bratteli diagrams (To appear, Colloquium Mathematicum) T Hamachi and M Osikawa, Ergodic groups of automorphisms and Krieger s theorems, Seminar on Math Keio Univ, 3 (98) [KS] M Keane and M Smorodinsky, A class of finitary codes, Israel JMath, 6(977), [KS] M Keane and M Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann of Math, 09 (979), [R] MK Roychowdhury, {m n }-odometer and the binary odometer are finitarily orbit equivalent, Contemporary Mathematics, AMS, Edited by I Assani, UNC Chapel Hill, 430 (007), 3-34)

20 0 M K ROYCHOWDHURY AND D J RUDOLPH [R] MK Roychowdhury, Irrational rotation of the circle and the binary odometer are finitarily orbit equivalent, Publ RIMS, Kyoto Univ, 43 (007), [RR] MK Roychowdhury and DJ Rudolph, Any two irreducible Markov chains of equal entropy are finitarily Kakutani equivalent (To appear, Isr J of Math)