Abstract This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli

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1 Abstract This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy. This answers a question of Ornstein and Weiss. 1

2 A measure-conjugacy invariant for free group actions Lewis Bowen January 17, 2008 Keywords: Ornstein s isomorphism theorem, Bernoulli shifts, measure conjugacy MSC:37A35 1 Introduction Let G be a countable group. Let (K, ν be a finite set and let ν be a probability measure on K. K G is the set of all functions x : G K. With the product topology, K G is homeomorphic to a Cantor set. G acts on K G by gx(f = x(g 1 f for x K G, g, f, G. Let ν G be the product measure on K G. The dynamical system (K G, ν G is called the Bernoulli shift over G with base measure ν. The entropy of the measure ν is defined by H(ν = k K ν({k} log(ν({k}. One consequence of A. Kolmogorov s seminal work [Ko58, Ko59] is that if two Bernoulli shifts (K1 Z, νz 1 and (KZ 2, νz 2 over the integers are measurably conjugate (i.e., metrically isomorphic then H(ν 1 = H(ν 2. This (and many related results was extended to a large class of amenable groups in [OW87]. At the end of that paper, D. Ornstein and B. Weiss ask whether any such results can be extended to nonamenable groups. Specifically, they asked the following. If G is a nonabelian free group and ν 1, ν 2 are the uniform probability measures on finite spaces K 1, K 2 with K 1 K 2 then is it possible that (K1 G, ν1 G is measurably conjugate to (K2 G, νg 2? To appreciate the difficulty of the question, let us recall how it was answered in the case G = Z. The central idea is the concept of entropy. If α = {A 1,..., A n } is a measurable partition of a probability space (X, µ, then the entropy of α is: n H(α = µ(a i log(µ(a i. i=1 The join of two partitions α and β is the partition α β = {A B A α, B β}. Now suppose G acts on a probability space (X, µ by measure-preserving transformations. If F i G is an increasing sequence of finite subsets of G, then the entropy rate of a partition α with respect to {F i } is: 1 ( h(α = lim i F i H gα. g F i 2

3 In this paper, we will always take F i to be the ball of radius i with respect to some fixed word metric on G. In the case G = Z, we take F i = [ i, i]. If G = s 1,..., s r is the free group of rank r then we let F i be the ball B(e, i of radius i centered at the identity element e with respect to the word metric associated to the generating set {s ±1 1,..., s±1 r }. The entropy h(x, µ of the system (X, µ is the supremum of h(α over all partitions α. In order to calculate this number, Y. Sinai proved that, if G = Z, then H(X, µ = h(α for any partition α that is generating; i.e., the smallest G-invariant σ-algebra containing α is, up to sets of measure zero, equal to the σ-algebra of all measurable subsets of (X, µ. An easy calculation now shows that, if G = Z, then the entropy of (K G, ν G is H(ν. Since entropy is clearly a measure-conjugacy invariant, this shows that if two Bernoulli shifts over Z are measurably conjugate then their base measures have the same entropy. Now let us consider the case when G = s 1,..., s r is a nonabelian free group. Fix n 2. Let u n be the uniform measure on K n = {1,..., n}. Let α = {A 1,..., A n } be the obvious partition of Kn G: A i is the set of all x Kn G such that x(e = i. Let α k := gα. g B(e,k A short calculation reveals that H(α k = B(e, k log(n so that h(α = log(n, as expected. To see unexpected behavior, observe that h(α 1 H(α n+1 = lim n B(e, n = lim B(e, n + 1 log(n n B(e, n = (2r 1 log(n. But α 1 and α are generating. So two different generating partitions may have different entropy rates. Even worse, since h(α m = (2r 1 m log(n, the supremum of h(β over all partitions β is positive infinity. Thus the systems (Kn G, ug n (n = 2, 3,... cannot be distinguished by these methods. One of the many useful characteristics that entropy possesses, in the case G = Z, is that it is monotone decreasing under factor maps; i.e. if φ : (X 1, µ 1 (X 2, µ 2 is a G-equivariant map such that φ µ 1 = µ 2, then h(x 1, µ 1 h(x 2, µ 2. This fact is due to Y. Sinai. For example, if n > m then there exists a factor map from (Kn Z, uz n onto (KZ m, uz m but there does not exist a factor map from (Km, Z u Z m onto (Kn, Z u Z n. In [OW87], D. Ornstein and B. Weiss exhibited a seemingly paradoxical example of a factor map from (K2 G, ug 2 onto (KG 4, ug 4 in the case in which G =< s 1, s 2 > is a rank 2 free group. Identify K 2 with Z/2Z and K 4 with Z/2Z Z/2Z. The factor map φ : (K2 G, u G 2 (K4 G, ug 4 is defined by φ(x(g = ( x(g + x(gs 1, x(g + x(gs 2. There is an obvious factor map from (K4 G, ug 4 (KG 2, ug 2 induced by a projection map Z/2Z Z/2Z Z/2Z. Thus, (K2 G, ug 2 and (KG 4, ug 4 are weakly isomorphic, i.e., each one factors onto the other. It is shown in 8 that, if G is any nonabelian free group, then all nontrivial Bernoulli shifts over G are weakly isomorphic. The main result of this paper is: 3

4 Theorem 1.1. Let G = s 1,..., s r be the free group of rank r. If (K 1, ν 1, (K 2, ν 2 are finite probability spaces and (K G 1, ν G 1 is measurably conjugate to (K G 2, ν G 2 then H(ν 1 = H(ν 2. The converse is also true. D. Ornstein famously proved that if H(ν 1 = H(ν 2 then (K1 Z, νz 1 is measurably conjugate to (KZ 2, νz 2 [Or70]. We will say that Ornstein s isomorphism theorem holds for a group G if H(ν 1 = H(ν 2 implies (K1 G, ν1 G is measurably conjugate to (K2 G, νg 2. A. M. Stepin [St75] applied Ornstein s result to obtain: Theorem 1.2 (St75. Let G be any countable group such that G contains a subgroup for which Ornstein s isomorphism theorem holds. Then Ornstein s isomorphism theorem also holds for G. Since free groups contain infinite cyclic subgroups and Ornstein s isomorphism theorem holds for Z, Stepin s result implies the converse of the theorem 1.1. To prove theorem 1.1, the following invariant is introduced. Let (X, µ be any probability space on which G = s 1,..., s r acts by measure-preserving transformations. For a partition α, let r F(α := (1 2rH(α + H(α s i α. i=1 f(α := lim n F(α n. This paper shows that the above limit exists (although it may equal and if α and β are any two generating partitions then f(α = f(β. This common number is therefore a measure-conjugacy invariant that we denote by f(x, µ. If (X, µ = (K G, ν G is Bernoulli, then theorem 5.3 below implies f(x, µ = H(ν. This proves theorem 1.1. The proof that f is a measure-conjugacy invariant consists of several steps. Any partition α, induces a natural topology, denoted here by Top(α, on X. If α and β are two partitions such that Top(α = Top(β then we say that α and β are topologically equivalent. The space of all finite partitions modulo measure zero sets is endowed with a canonical metric. This paper proves that if α is generating then the set of partitions topologically equivalent to α is dense in the subspace of all generating partitions. As far as I know, this is a new result even in the case G = Z. Moreover, any two topologically equivalent generating partitions have a common splitting. To prove that f is an invariant, it is shown that F is continuous on the space of partitions (this is obvious and that F is monotone decreasing under splittings. Section 6 contains examples of f(x, µ calculations. For instance, if G = Z, then f equals the entropy rate h. Section 7 contains a proof of Stepin s theorem 1.2. It is based on co-induced actions and Ornstein s theorem. The full details of Stepin s original proof have not appeared but it seems likely that his proof is similar to the one presented here. Section 8 contains a proof that all nontrivial Bernoulli shifts over a fixed nonabelian free group are weakly isomorphic. Organization In 2, some standard definitions and results are presented. In 3, α-topological partitions are defined and it is proven that, if α is an arbitrary generating partition, then the set of 4

5 partitions topologically equivalent to α is dense in the subspace of all generating partitions. In 4, splittings are defined and it is proven that any two topologically equivalent generating partitions have a common splitting. In 5, it is proven that F is monotone decreasing under splittings and f(k G, ν G = H(ν. This concludes the proof of theorem 1.1. In 6, examples of f(x, µ calculations are presented. In 7, theorem 1.2 is proven. In 8, it is proven that all nontrivial Bernoulli shifts over a fixed nonabelian free group are weakly isomorphic. Acknowledgments: I am grateful to Russell Lyons for suggesting this problem, for encouragements and for many helpful conversations. I would like to thank Dan Rudolph for pointing out errors in a preliminary version. I would like to thank Benjy Weiss for many comments that helped me to greatly improve the exposition of this paper and simplify several arguments. 2 Some Standard Definitions Definition 1. Let G be a group acting on probability spaces (X 1, µ 1 and (X 2, µ 2 by measure-preserving transformations. A measurable map φ : X 1 X 2 (X i X i for i = 1, 2 is a measure-conjugacy if µ i (X i = 1 (i = 1, 2, φ is G-equivariant (i.e., φ(gx = gφ(x for all g G, x X 1, φ µ 1 = µ 2 and φ is invertible. A measure-conjugacy is also called a metric isomorphism. For the rest of this section, fix a probability space (X, µ. Definition 2. A partition α = {A 1,..., A n } is a pairwise disjoint collection of measurable subsets A i of X such that n i=1a i = X. The sets A i are called the partition elements of α. Alternatively, they are called the atoms of α. Unless stated otherwise, all partitions in this paper are finite (i.e. n <. Let P 0 denote the set of all finite partitions. If G is a group acting on (X, µ then G acts on P 0 by g{a 1,..., A n } = {ga 1, ga 2,..., ga n } for all g G and {A 1,..., A n } P 0. Definition 3. If α, β P 0 then the join of α and β is the partition α β = {A B A α, B β}. Definition 4. The information function I(α : X R corresponding to a partition α = {A 1,..., A n } is defined by where A j is the atom of α containing x. I(α(x = log(µ(a j Definition 5. The entropy H(α of α = {A 1,..., A n } is defined by where we set 0 log(0 = 0. H(α = n µ(a i log(µ(a i = i=1 5 x X I(x dµ(x

6 Definition 6. Let F be a σ-algebra contained in the σ-algebra of all measurable subsets of X. Given a partition α, define the conditional information function I(α F : X R by I(α F(x = log ( µ(a j F(x where, as above, A j is the atom of α containing x. Here µ(a j F : X R is the conditional expectation of χ Aj, the characteristic function of A j, with respect to the σ-algebra F. In other words, it is the unique F-measurable function such that for all F-measurable functions f : X R, µ(a j F(xf(x dµ(x = χ Aj (xf(x dµ(x. X The conditional entropy of α with respect to F is defined by H(α F = I(α F(x dµ(x. X If β is a partition then, by abuse of notation, we can identify β with the σ-algebra equal to the set of all unions of partition elements of β. Through this identification, I(α β and H(α β are well-defined. Lemma 2.1. For any two partitions α, β and for any two σ-algebras F 1, F 2 with F 1 F 2, H(α β = H(α + H(β α, X H(α F 2 H(α F 1. Proof. This is well-known. For example, see [Gl03, Proposition 14.16, page 255]. Definition 7. For any sets A, B let A B denote the symmetric difference of A and B. I.e., A B = (A B (B A. Definition 8. Two partitions α and β agree up to measure zero if, after reordering their atoms if necessary, α = {A 1,..., A n }, β = {B 1,..., B m }, and there is some number 1 r min(m, n such that 1. µ(a i B i = 0 for all i with 1 i r, 2. µ(a i = 0 for all i > r, 3. µ(b i = 0 for all i > r. Let P = P 0 / where is the equivalence relation α β if α and β agree up to measure zero. All of the concepts we introduce and use (e.g., entropy and conditional entropy only depend on the equivalence class of a partition α P 0. So, by abuse of notation, we will often refer to elements of P as if they are partitions. 6

7 Definition 9. Define d : P P R by d(α, β = H(α β + H(β α = 2H(α β H(α H(β. By [Pa69, theorem 5.22, page 62] this defines a distance function on P. If G is a group acting by measure-preserving transformations on (X, µ then the action of G on P is isometric. I.e., if g G, α, β P then d(gα, gβ = d(α, β. Lemma 2.2. Suppose, for each i 1, α i = {A i 1,..., Ai n i } is a sequence of partitions, β = {B 1,..., B m } P and for all j = 1...m, lim i µ(a i j B j = 0, n i < C for some constant C and for all i 1. Then the sequence {α i } converges to β with respect to the metric defined in the previous definition. Proof. This is an elementary exercise. Definition 10. Let G be a group acting on (X, µ. Let α be a partition. Let Σ α be the smallest G-invariant σ-algebra containing the atoms of α. Then α is generating if for all measurable sets A X there exists a set A Σ α such that µ(a A = 0. Let P gen P denote the set of all generating partitions. 3 Topologically Equivalent Partitions For this section, fix a countable group G and an action of G on a probability space (X, µ by measure-preserving transformations. Definition 11. Given a partition α = {A 1,..., A n }, there is a natural topology Top(α on X induced by α defined as follows. Let K = {1,..., n}. Let K G denote the set of all functions x : G K with the product topology. Define N α : X K G by N α (x(g = j if g 1 x A j. This map is measurable since the partition elements of α are measurable. Let Top(α denote the smallest topology on X such that N α is continuous (i.e., a set O X is open if and only if O = Nα 1 (O for some open subset O K G. If α is generating, then N α is an embedding so that (X, Top(α is metrizable and µ is Borel. Definition 12. A partition β = {B 1,..., B m } P is α-topological if each partition element B i is clopen in Top(α. Let P(α P denote the set of α-topological partitions. Definition 13. Two partitions α, β are topologically equivalent if α P(β and β P(α, i.e., Top(α = Top(β. Let P eq (α P(α denote the subspace of partitions topologically equivalent to α. 7

8 The goal of this section is to prove theorem 3.4 below: if α is a generating partition then P eq (α is dense in the subspace of all generating partitions. Definition 14. Let n > 1. Let K n = {1,..., n}. Let x K G n. Let F G be a finite subset of G. The cylinder set associated to x and F is defined by Cyl(x, F = {y K G n y(g = x(g g F }. An α-cylinder set is a set of the form N 1 α (Cyl(x, F X where N α is as defined in definition 11. Lemma 3.1. For any n > 1, every clopen subset of K G n is a finite union of cylinder sets. Proof. Let C Kn G be clopen. The cylinder sets form a basis for the topology of open sets in Kn G. So C is a union of cylinder sets. Let G 0 G 1... be an increasing sequence of finite subsets of G such that i=0g i = G. A cylinder set Cyl(x, F has order m if F G m and F G m 1. If C is not a finite union of cylinder sets then there exists an increasing infinite sequence {m i } i=1 such that for all i 0 there exists a cylinder set Cyl(x i, F i of order m i satisfying Cyl(x i, F i C, if Cyl(y, F is a cylinder set such that Cyl(x i, F i Cyl(y, F C then Cyl(y, F has order at least m i. But then every limit point of the sequence {x i } is not contained in C. So C is not closed. This contradiction implies the lemma. Lemma 3.2. Let α be a generating partition and β = {B 1,..., B m } P. Let ǫ > 0. Then there exists an α-topological partition β = {B 1,..., B m } such that for all i = 1...m, µ(b i B i < ǫ. Proof. Let X have the topology Top(α. Since α is generating, µ is Borel. So the Riesz representation theorem implies that if B X is measurable then µ(b = inf O µ(o where the infimum is over all open subsets O containing B. Since every open set is a union of α-cylinder sets, this implies, for every δ > 0 there exists a clopen set B with µ(b B < δ. For each i, let B i X be a clopen set with µ(b i B i < δ := ǫ. For i = 1...m 1, let m Observe that for all i = 1...m, B m = X B i = B i B j. j i i<mb i = B m i j B i B j. B i j B j B j B i B i j B j B j. 8

9 Thus µ(b i B i mδ = ǫ. By construction, β = {B 1,..., B m } is a clopen partition and therefore α-topological. Lemma 3.3. Let α, β P gen. Let α = {A 1,..., A n }. For each k 0, let β k := g B(e,k Let ǫ > 0. Then there exists K such that for all k K the partition elements {B k 1,..., B k m k } of β k can be ordered so that there exists an r {1,..., m k } and a function f : {1, 2,...r} {1, 2,..., n} so that for all i {1,..., r}, gβ. µ(b k i A f(i µ(b k i 1 ǫ and ( µ i>r B k i < ǫ. ( ǫ 2. Proof. Let δ > 0 be such that δ < n By the previous lemma, there exists a partition α = {A 1,..., A n} P(β such that µ(a i A i < δ for all i. Since each A i is clopen (with respect to Top(β, each is a finite union of β-cylinder sets (by lemma 3.1. Therefore, there exists a K such that for all k > K, each atom A i of α is a finite union of atoms of β k. Equivalently, the partition α is refined by β k. Let β k = {B1 k,..., Bk m k } and let f : {1,..., m k } {1,..., n} be the function f(i = j if Bi k A j. This is well-defined since β k refines α. After reordering the partition elements of β k = {B1 k,..., Bk m k } if necessary, we may assume that there is an r {0,..., m k } such that, if r > 0 then for all i r, µ(b k i A f(i µ(b k i 1 δ, and if i > r then So if i > r then So µ(b k i A f(i µ(b k i < 1 δ. µ(b k i A f(i < (1 δµ(b k i. µ(b k i = µ(bk i A f(i + µ(b k i A f(i < µ(b k i A f(i + (1 δµ(b k i. 9

10 Solve for µ(bi k to obtain µ(b k i < 1 δ µ(b k i A f(i. Since the atoms B k i are pairwise disjoint, it follows that ( µ i>r B k i < 1 ( µ Bi k A f(i. δ Since B k i A f(i, it must be that Bk i A f(i A f(i A f(i. So, i>r ( µ i>r B k i 1 ( µ A f(i A f(i δ i>r n δ < ǫ. Theorem 3.4. If α is a generating partition then P gen P eq (α. I.e., the subspace of partitions topologically equivalent to α is dense in the space of all generating partitions. Proof. For this proof, let X have the topology Top(α. Let α = {A 1,..., A n } and β = {B 1,..., B m } P gen. Without loss of generality, we assume that µ(a i > 0 for all i = 1...n. Let ǫ > 0. By the previous lemma, there exists a K 0 such that for any k K, the atoms of β k = {B k 1,..., Bk m k } can be ordered so that there exists an r {1,..., m k } and a function f : {1, 2,...r} {1, 2,..., n} so that for all i {1,..., r}, µ(b k i A f(i µ(b k i 1 ǫ and ( µ i>r B k i < ǫ. (1 By choosing ǫ small enough (if necessary we may assume that f is onto (for example, by choosing ǫ to be smaller than 1 2 µ(a j over all j = 1...n. By definition of β k, m k m B(e,k. If necessary, we may assume that m k = m B(e,k after modifying β k by adding to it several copies of the empty set. That is, for some i, it may occur that B k i =. Let δ > 0 be such that δ < ǫ. By lemma 3.2 there exists an α-topological partition γ = {C 1,..., C m } such that µ(c i B i < δ for all i. By choosing δ small enough we may 10

11 assume the following. Let γ k = {C k 1,..., Ck m k }. Then, after reordering the atoms of γ k if necessary, Let C i ( m k µ j=1 Cj k Bk j ǫ. (2 = {x C i if x C k j for some j then x A f(j} = m k j=1 Let C i,j = C i A j C i. Let C i C k j A f(j. γ 1 = {C i i = 1...m} {C i,j 1 i, j m}. Observe that C i and C i,j are clopen with respect to Top(α. So γ 1 P(α. We claim that γ 1 is topologically equivalent to α. Let Σ 1 be the smallest G-invariant collection of subsets of X that is closed under finite intersections and unions and contains the atoms of γ 1. It suffices to show that every atom of α is in Σ 1. Observe that, for each i, C i = C i m j=1 C i,j. Hence, C i Σ 1. Therefore the atoms of γ k are also in Σ 1. Since f is onto, the definition of C i implies C i A p = {C i Ck j f(j = p}. So C i A p is in Σ 1 for all i, p. Now C i A p = C i,p (C i A p. So C i A p Σ 1 for all i, p. Since m A p = C i A p, A p Σ 1. Since p is arbitrary, this proves the claim. Thus γ 1 P eq (α. We claim that µ(c i C i 3ǫ for all i. By definition, For each j, Thus, i=1 m k C i C i = C i C i j=1 C k j A f(j. C k j A f(j (C k j Bk j (Bk j A f(j. m k r C i C i (Cj k Bj k j j=1(b k A f(j (Bj k A f(j. (3 j>r j=1 If j r, then by definition of r, µ(b k j A f(j µ(b k j 1 ǫ. 11

12 This implies µ(b k j Ak f(j ǫµ(bk j. Thus ( r µ j=1 Bj k Ak f(j j ǫµ(bj k ǫ. (4 Equations 3, 2, 4 and 1 imply the claim. Since δ < ǫ and µ(c i B i < δ for all i, the above claim implies that µ(c i B i 4ǫ for all i. Thus we have shown that for every ǫ > 0, there exists a partition γ 1 = {C 1,..., C m,...}, topologically equivalent to α, containing at most m + m 2 partition elements and such that µ(c i B i < 4ǫ for i = 1...m. Lemma 2.2 now implies that β is in the closure of P eq (α. Since β is arbitrary this implies the theorem. 4 Splittings In this section, G can be any finitely generated group with finite symmetric generating set S. Definition 15. Let α and β be partitions. If, for every atom A α there exists an atom B β such that A B then we say α refines β. Equivalently, β is a coarsening of α. Definition 16. Let α be a partition. A simple splitting of α is a partition σ of the form σ = α sβ where s S and β is a coarsening of α. A splitting of α is any partition σ that can be obtained from α by a sequence of simple splittings. In other words, there exist partitions α 0, α 1,..., α m such that α 0 = α, α m = σ and α i+1 is a simple splitting of α i for all 1 i < m. If σ is a splitting of α then α and σ are topologically equivalent. Definition 17. The Cayley graph Γ of (G, S is defined as follows. The vertex set of Γ is G. For every s S and every g G there is a directed edge from g to gs labeled s. There are no other edges. The induced subgraph of a subset F G is the largest subgraph of Γ with vertex set F. A subset F G is connected if its induced subgraph in Γ is connected. Lemma 4.1. If α, β P, α refines β and F G is finite, connected and contains the identity element e then α is a splitting of α. f F 1 fβ 12

13 Proof. We prove this by induction on F. If F = 1 then F = {e} and the statement is trivial. Let f 0 F {e} be such that F 1 = F {f 0 } is connected. To see that such an f 0 exists, choose a spanning tree for the induced subgraph of F. Let f 0 be any leaf of this tree that is not equal to e. By induction, α 1 := α f F 1 1 fβ is a splitting of α. Since F is connected, there exists an element f 1 F 1 and an element s 1 S such that f 1 s 1 = f 0. Since f 1 F 1, α 1 refines (f1 1 β. Thus α fβ = α 1 f0 1 β = α 1 s (f1 β f F 1 is a splitting of α. Lemma 4.2. If α, β P and β P(α then there exists a number k 0 such that α k = g B(e,k gα refines β. Proof. This follows from lemma 3.1. Proposition 4.3. Let α, β be two topologically equivalent generating partitions. Then there is an n 0 such that α n = gα g B(e,n is a splitting of β. Of course, α n is also a splitting of α. This proposition is a variation of a result that is well-known in the case G = Z in the context of subshifts of finite-type. For example, see [LM95, theorem 7.1.2, page 218]. It was first proven in [Wi73]. Proof. After replacing β with β k for some sufficiently large k, we may assume, by the previous lemma, that β refines α. The previous lemma now implies that for some n 0, α n refines β. By the previous lemma, β α n is a splitting of β. Since α n refines β, β α n = α n. Theorem 4.4. Let F : P R be any continuous function. Suppose that, whenever σ is a splitting of a partition α then F(σ F(α. Define f : P R by f(α = lim n F(α n = inf n F(αn. Then, for any two generating partitions α 1 and α 2, f(α 1 = f(α 2. So we may define f(x, µ = f(α for any generating partition α. Then f(x, µ is a measure-conjugacy invariant of the system (X, µ, i.e., if (Y, ν is any system that is measurably-conjugate to (X, µ then f(x, µ = f(y, ν. Proof. Let α and β be two topologically equivalent partitions. We claim that f(α = f(β. To see this, suppose, for a contradiction, that f(α < f(β. Then there exists an n 0 such that F(α n < f(β. But by the previous proposition, there is an m 0 such that β m is a splitting of α n which implies F(α n F(β m f(β, a contradiction. So f(α = f(β. 13

14 For n 0 and α P, let F n (α = F(α n. Since F is continuous and the map α α n is also continuous, it follows that F n is continuous. Since f(α = inf n F n (α, it follows that f is upper semi-continuous, i.e., if {β n } is a sequence of partitions converging to α then lim sup n f(β n f(α. Now let α, β be two generating partitions. By theorem 3.4, there exist partitions {β n } n=1 topologically equivalent to β such that {β n } n=1 converges to α. So f(β = lim sup n f(β n f(α. Similarly, f(α f(β. So f(α = f(β. We will show in the next section that there is a function F for which the related invariant f as defined above is computable and nontrivial. 5 The f-invariant In this section, G denotes a finitely generated group with symmetric generating set S. Some of the results of this section hold for arbitrary groups but most of them are only interesting in the special case in which G is a finitely generated free group. So we will assume that G = s 1,..., s r and that S = {s ±1 1,..., s ±1 r }. Note S = 2r. Definition 18. Let α P. Define F(α := 1 ( 2 s S H(α sα ( S 1H(α. Because µ is G-invariant, H(α sα = H(s 1 α α = H(α s 1 α. Since S = 2r, the above definition is equivalent to the definition given in the introduction. Definition 19. Suppose σ = α tβ is a simple splitting of a partition α where t S and β is a coarsening of α. σ is a simple independent splitting if H(σ α sσ = H(σ α sα = H(σ α for all s S {t}. If σ is any, perhaps nonsimple, splitting of α, we will say that σ is an independent splitting of α if there exist partitions α 0, α 1,..., α m such that α 0 = α, α m = σ and α i+1 is a simple independent splitting of α i for all 1 i < m. Theorem 5.1. Let α P. If σ is a splitting of α then F(σ F(α. Equality holds if and only if σ is an independent splitting of α. Proof. It suffices to prove the proposition in the special case in which σ is a simple splitting of α. By definition, σ = α tβ for some t S and partition β that α refines. By lemma 2.1, for any s S, H(σ sσ = H(α sα + H(σ sσ α sα = H(α sα + H(sσ α sα + H(σ α sα sσ H(α sα + H(σ α s 1 α + H(σ α sα. 14

15 The last inequality occurs because µ is G-invariant, so H(sσ α sα = H(σ α s 1 α. Also sα sσ = sσ refines sα, so H(σ α sα sσ H(σ α sα. Because α tα refines σ, H(σ α tα = 0. By lemma 2.1 again, H(σ = H(α+H(σ α. So, F(σ = ( 1 2 s S ( 1 2 s S = F(α + H(σ sσ ( S 1H(σ H(α sα + H(σ α sα + H(σ α s 1 α ( S 1 ( H(α + H(σ α s S {t} H(σ α sα H(σ α. It follows from lemma 2.1 that for all s S {t}, H(σ α sα H(σ α 0. Thus F(σ F(α. It is now easy to check that F(σ = F(α if and only if the splitting is independent. Definition 20. For any partition α, let α k = g B(e,k gα and define f(α := lim k F(α k = inf k F(αk. Corollary 5.2. If α 1, α 2 are any two generating partitions, then f(α 1 = f(α 2. Thus we may define f(x, µ = f(α for any generating partition α. f(x, µ is a measure-conjugacy invariant. Proof. This follows from the above theorem and theorem 4.4. Remark 1. The above definition of f makes sense even if G is not a free group. But, if S = {s ±1 1,..., s±1 r } then there is an obvious surjective homomorphism from the free group s 1,..., s r onto G. Through this homomorphism, s 1,..., s r acts on (X, µ. The f-invariant is the same regardless of whether the acting group is G or s 1,..., s r. Therefore, we might as well assume that G = s 1,..., s r. Definition 21. Let K be a finite set and ν a probability measure on K. Let K G be the product space with the product measure ν G. The space (K G, ν G is called the Bernoulli shift over G with base measure ν. Let A k = {x K G x(e = k} where e denotes the identity element in G. Then α = {A k k K} is called the Bernoulli partition associated to K. It is generating and H(ν = H(α, by definition. Theorem 5.3. Let G = s 1,..., s r be the free group of rank r. Let K be a finite set and ν a probability measure on K. Then f(k G, ν G = H(ν. 15

16 Proof. Let α be the Bernoulli partition associated to K. Let g 1,.., g n be n distinct elements of G. It follows from the Bernoulli condition that the collection {g i α} n i=1 of partitions is independent. This means that if j : {1,..., n} K is any function then ν G ( n i=1 g i A j(i = n i=1 ν G (A j(i. It is well-known that this implies ( n H i=1 g i α = n H(g i α = nh(α. See, for example, [Gl03, prop , page 257]. So for any k 1, F(α k = = ( 1 2 s S ( 1 2 s S i=1 H(α k sα k ( S 1H(α k B(e, k B(s, k H(α ( S 1 B(e, k H(α. Suppose r > 1. Then, since G = s 1,..., s r is free, it is a short exercise to compute: for all s S. Thus, B(e, k = 1 + S ( S 1k 1, S 2 B(e, k B(s, k = 2 ( S 1k+1 1 S 2 ( F(α k = H(α S ( S 1k+1 1 ( S 1 ( S 1 S ( S 1k 1 S 2 S 2 = H(α. If r = 1 then B(e, k = 2k + 1 and B(e, k B(s, k = 2k + 2. So it follows in a similar way that F(α k = H(α. So f(x, µ = lim k F(α k = H(α = H(ν. Theorem 1.1 is an immediate consequence of the previous two results. 6 Examples In this section, the following is proven. Theorem 6.1. Let G = s 1,..., s r be the free group on r generators. Let G act on a probability space (X, µ by measure-preserving transformations. 16

17 1. If G = Z then f(x, µ = h(x, µ, the Kolmogorov entropy of (X, µ. 2. If X has finite cardinality then f(x, µ = (1 rh(µ. 3. If there is an isomorphism (X, µ = (X 1, µ 1 (X 2, µ 2 where (X i, µ i are probability spaces on which G acts by measure-preserving transformations and the action of G on (X, µ is the product action then, f(x, µ = f(x 1, µ 1 + f(x 2, µ Suppose φ : G G 2 is a surjective homomorphism onto an infinite group G 2. For g G let B 2 (g, n denote the ball of radius n in G 2 centered at φ(g with respect to the word metric induced by the generating set φ(s. Suppose G 2 satisfies the following growth condition: B 2 (e, n + 1 lim inf < 2 S 1. n B 2 (e, n S Let (X, µ = (K G 2, ν G 2 for some finite probability space (K, ν with H(ν > 0. If the action of G on X is given by gx = φ(gx, where G 2 acts on K G 2 in the usual way, then f(x, µ =. Remark 2. Statement 2 easily implies the existence of spaces (X, µ for which f(x, µ < 0 in the case r > 1. This contrasts with the fact implied by statement 1 that, if r = 1, then f(x, µ = h(x, µ 0. Statements 2 and 3 are well-known in the case r = 1. The idea underlying statement 4 was communicated to me by Russell Lyons [Ly06]. Statements 2 and 4 lead to the following question. If the action of G on (X, µ is strongly mixing then is f(x, µ 0? We will prove each statement above in separate lemmas. Lemma 6.2. Statement 1 of theorem 6.1 is true. Proof. Let G = Z and S = {s 1, s}. Since µ is G-invariant, H(α sα = H(α s 1 α. So F(α = H(α sα H(α. By corollary 5.2, f(α = lim n F(α n where α n = g B(e,n gα. Thus, f(α = lim F(α n n = lim H( n+1 i= n si α H( n n i= ns i α = lim H(s n+1 α n i= n s i α n = H(α 1 i= si α. The third equality follows from lemma 2.1. The last one follows from G-invariance of µ. It is well-known that the last limit above equals h(α, the entropy rate of α (e.g., it is in [Gl03, theorem 14.29, page 262]. Lemma 6.3. Statement 2 of theorem 6.1 is true. 17

18 Proof. We may assume that for all x X, µ({x} > 0. If α is any generating partition, then for all n large enough, α n = {{x} : x X}. Thus H(α n = H(µ. Similarly, H(α n sα n = H(µ for all s S. The formula now follows from the definitions of F and f. Lemma 6.4. Statement 3 of theorem 6.1 is true. Proof. For i = 1, 2, let α i be a generating partition for (X i, µ i. Then α := α 1 α 2 = {A 1 A 2 : A 1 α 1, A 2 α 2 } is a generating partition for (X, µ. Moreover, α n = α n 1 αn 2. It is well-known that H(α n = H(α n 1 + H(αn 2. Similarly, H(αn sα n = H(α n 1 sαn 1 + H(α n 2 sα n 2 for all s S. The statement now follows from the definitions of F and f. Lemma 6.5. Statement 4 of theorem 6.1 is true. Proof. Let α be the canonical generating partition of (K G 2, ν G 2 : α = {A k k K} where A k = {x K G 2 x(e = k}. The proof of theorem 5.3 shows that ( 1 F(α n = B 2 (e, n B 2 (s, n H(α ( S 1 B 2 (e, n H(α. 2 s S Since B 2 (e, n B 2 (s, n B 2 (e, n + 1, F(α n H(α B 2(e, n S ( B2 (e, n + 1 ( S B 2 (e, n S So f(x, µ = lim inf n F(α n =. 7 Stepin s theorem from co-induced actions In this section, theorem 1.2 is proven. First, we need to discuss co-induced actions. These actions have been used in [Da06] in an investigation of spectral properties of ergodic actions of discrete groups, in [Ga05] in orbit equivalence theory and in [DGRS07] in constructing non-bernoulli CPE actions of amenable groups. The definition is related to but different from the well-known Mackey-Zimmer definition of an induced action [Zi78, Zi84]. Fix a countable group G and a subgroup H < G. Definition 22. A section for G/H is a map σ : G/H G such that σ(gh gh for all g G. Fix such a section σ. For convenience, we assume σ(h = e. Let α : G G/H H be the cocycle α(g, c = σ(c 1 gσ(g 1 c g G, c G/H. Let (W, ω be a probability space on which H acts by measure-preserving transformations. Define an action of G on the product space (W G/H, ω G/H by (gx(c = α(g, cx(g 1 c g G, x W G/H, c G/H. 18

19 A routine calculation shows that (g 1 g 2 x(c = (g 1 (g 2 x(c for all g 1, g 2 G, x W G/H, c G/H so that this action is well-defined. The system (W G/H, ω G/H is said to be co-induced from the H-action on (W, ω. Since H preserves ω and α(g, c H for all g G, c G/H, it follows that G acts on (W G/H, ω G/H by measure-preserving transformations. Definition 23. Let (W 1, ω 1, (W 2, ω 2 be probability spaces on which H acts by measurepreserving transformations. Suppose φ : W 1 W 2 is a factor map. Define Φ : W G/H 1 W G/H 2 by Φ(x(c = φ(x(c, x W G/H 1, c G/H. Φ is the factor map induced by φ. To check G-equivariance, note: Φ(gx(c = φ(gx(c = φ(α(g, cx(g 1 c = α(g, cφ(x(g 1 c = α(g, cφ(x(g 1 c = ( gφ(x (c. It is easy to check that Φ ω 1 = ω 2. If φ is invertible then so is Φ with inverse defined by: We can now prove theorem 1.2. Φ 1 (x(c = φ 1 (x(c, x W G/H 2, c G/H. Proof of theorem 1.2. Let H be a subgroup of G for which Ornstein s isomorphism theorem holds. So there exists a measure-conjugacy φ : (K1 H, ν1 H (K2 H, ν2 H. By the previous definition, the induced map Φ : ( ((K1 H G/H, (ν H 1 G/H ( ((K2 H G/H, (ν H 2 G/H is a measure-conjugacy. Now, it suffices to show that for each i = 1, 2, the map J : ( ( ((Ki Ki G, νi G H G/H, (ν H i G/H defined by J(x(c(h = x(σ(ch, x K G i, c G/H, h H is a measure-conjugacy. To check that J is G-equivariant, note that for g G, x K G i and c G/H, J(gx(c is the map h (gx(σ(ch = x(g 1 σ(ch. On the other hand, (gj(x(c = α(g, cj(x(g 1 c. Since J(x(g 1 c is the map h x(σ(g 1 ch, (gj(x(c is the map h x(σ(g 1 cα(g, c 1 h = x(g 1 σ(ch. Thus J(gx = gj(x. We leave it to the reader to check that J ν G i J 1 (y(g = y(gh(α(g, gh, y (K H i G/H, g G. = (ν H i G/H and 19

20 8 Weak Isomorphisms If (X, µ is a probability space that is not measurably isomorphic to a finite probability space, then let H(µ = +. In this section, we prove: Theorem 8.1. Let (X 1, ν 1, (X 2, ν 2 be probability spaces with H(ν i (0, ] for i = 1, 2. Let G be a nonabelian free group. Then (X G 1, νg 1 is weakly isomorphic to (XG 2, νg 2. For the remainder of this section, G =< s 1,.., s r > is a fixed nonabelian free group. Lemma 8.2. If H(ν 1 H(ν 2 then there exists a factor map Φ : (X G 2, νg 2 (XG 1, νg 1. Proof. By Ya. Sinai s factor theorem [Si62] (also in [Gl03, theorem 20.13, page 360], there exists a factor map φ : (K2 Z, νz 2 (KZ 1, νz 1. By definition 23, the coinduced map Φ is a factor map from (K2 G, νg 2 onto (KG 1, νg 1. Here we have identified Z with a subgroup of G. For n N, let K n = {1,..., n} and u n be the uniform probability measure on K n. Lemma 8.3. There exists a factor map φ : (K G 2, ug 2 (KG 4, ug 4. Proof. As in the Ornstein-Weiss example given in the introduction, identify K 2 with Z/2Z and K 4 and Z/2Z Z/2Z. For x K2 G and g G, define φ(x(g = ( x(g + x(gs 1, x(g + x(gs 2. We leave it to the reader to check that φ is the required factor map. Let K = (Z/2Z N and let u be the product measure u N 2 on K. The next lemma is due to Adam Tímar. It was used in [Ba05]. Lemma 8.4. There exists a factor map Φ : (K G 2, ug 2 (KG, ug. Proof. For each n 0, identify K 2 n with (Z/2Z n. Let φ : (K2 G, ug 2 (KG 4, ug 4 be defined as in the previous lemma. Consider the product map: φ n : K G 2 n KG 2 K G 2 n KG 4 defined by φ n (x, y = (x, φ(y. We can identify K 2 n+1 with K 2 n K 2 by the map (i 1,..., i n+1 ( (i 1,..., i n, i n+1. By taking the product, we can now identify K G 2 n+1 with K G 2 n KG 2. Similarly, we can identify K G 2 n+2 with K G 2 n KG 4. Therefore, we may regard φ n as a factor map from K G 2 n+1 to K G 2 n+2. Let Φ n : K G 2 K G 2 n+2 be the composition Φ n := φ n... φ 1 φ. For 1 m < n define π m : (Z/2Z n Z/2Z, π m (i 1,..., i n = i m. 20

21 Note that if n 1, n 2 > m, x K2 G and g G then ( π m Φn1 (x(g ( = π m Φn2 (x(g. Therefore, we may define Φ : (K2 G, ug 2 (KG, ug ( by π m Φ(x(g = Φm (x(g, m N, x K2 G, g G. Φ is the required factor map. The construction in the following lemma was communicated to me by Benjy Weiss [We08]. Lemma 8.5. Let (K, ν be a finite probability space with H(ν > 0. Then there exists a factor map from (K G, ν G onto (K G, ug. Proof. Suppose H(ν log(2. By lemma 8.2, (K G, ν G factors onto (K2 G, ug 2 which factors onto (K, G u G by the previous lemma. Now suppose that q = H(ν (0, log(2]. By Stepin s theorem 1.2, (K G, ν G is measurablyconjugate to (X G, µ G where X is the three-element space X = {0, 1, } and µ({0} = µ({1} = p for some p > 0. We identify the subset {0, 1} X with the group Z/2Z. Let Y be the disjoint union of Z/2Z Z/2Z with { }. Define φ : X G Y G as follows. For x X G and g G, if x(g =, then set φ(x(g :=. Otherwise let φ(x(g = ( x(g + x(gs k 1, x(g + x(gsl 2 Z/2Z Z/2Z where k, l 0 are the smallest positive integers such that x(gs k 1 Z/2Z and x(gs l 2 Z/2Z. It can be checked that φ is a factor map from (X G, µ G onto (Y G, λ G where λ is defined by λ({z} = p/2 for all z Z/2Z Z/2Z and λ({ } = µ({ } = 1 2p. So H(λ = H(µ + 2p log(2 = H(ν + 2p log(2. So we have shown that if H(ν (0, log(2] then (K G, ν G factors onto the Bernoulli shift with base measure entropy H(ν + 2p log(2 where p is defined by H(ν = 2p log(p (1 2p log(1 2p. Note that p is an increasing function of H(ν. So by composing factor maps together, we have that (K G, ν G factors onto the Bernoulli shift with base measure where p 1,..., p n satisfy H n := H(ν + 2p log(2 + 2p 1 log(2 + 2p 2 log( p n log(2 H(ν + 2p log( p i log(2 = 2p i+1 log(p i+1 (1 2p i+1 log(1 2p i+1 for i {1,..., n 1}. If we set H 0 = H(ν then n can be chosen to be any positive integer such that H n 1 log(2. Since p i+1 p i > 0 for all i, we have that, for n large enough, H n log(2. Thus, (K G, ν G factors onto a Bernoulli shift with base measure entropy at least log(2. By the first paragraph, all such shifts factor onto (K G, u G. So, by composing, (K G, ν G factors onto (K G, ug too. 21

22 Proof of theorem 8.1. By symmetry, it suffices to show that (X1 G, νg 1 admits a factor map onto (X2 G, ν2 G. By the previous lemma, (X1 G, ν1 G factors onto (K, G u G. By lemma 8.2, (K G, ug factors onto (XG 2, νg 2. The composition of these factor maps is the required factor map. References [Ka05] K. Ball. Factors of independent and identically distributed processes with nonamenable group actions. Ergodic Theory Dynam. Systems 25 (2005, no. 3, [Da06] A. I. Danilenko. Explicit solution of Rokhlin s problem on homogeneous spectrum and applications. Ergodic Theory Dynam. Systems 26 (2006, no. 5, [DGRS07] A.H. Dooley, V.Ya. Golodets, D. Rudolph and S.D. Sinel shchikov. Non- Bernoullian systems with completely positive entropy for a class of amenable groups, Ergodic Theory and Dynamical Systems (2007 (online. [Ga05] D. Gaboriau. Examples of groups that are measure equivalent to the free group. Ergodic Theory Dynam. Systems 25 (2005, no. 6, [Gl03] E. Glasner. Ergodic theory via joinings. Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, xii+384 pp. [Ko58] A. N. Kolmogorov. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. (Russian Dokl. Akad. Nauk SSSR (N.S [Ko59] A. N. Kolmogorov. Entropy per unit time as a metric invariant of automorphisms. (Russian Dokl. Akad. Nauk SSSR [LM95] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, xvi+495 pp. [Ly06] R. Lyons. personal communication. [Or70] D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Advances in Math [OW87] D. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math. 48 (1987, [Pa69] W. Parry. Entropy and generators in ergodic theory. W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+124 pp. [Si62] Ya. G. Sinaĭ. A weak isomorphism of transformations with invariant measure. (Russian Dokl. Akad. Nauk SSSR

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