ANNALS OF MATHEMATICS
A measure-conjugacy invariant for free group actions
By Lewis Phylip Bowen
SECOND SERIES, VOL. 171, NO. 2 March, 2010
anmaah
Annals of Mathematics, 171 (2010), 1387–1400
A measure-conjugacy invariant for free group actions
By LEWIS PHYLIP BOWEN
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. Introduction
This paper is motivated by an old and central problem in measurable dynamics: given two dynamical systems, determine whether they are measurably-conjugate, i.e., isomorphic. Let us set some notation. A dynamical system (or system for short) is a triple .G; X; / where .X; / is a probability space and G is a group acting by measure-preserving transformations on .X; /. We will also call this a dynamical system over G, a G-system or an action of G. In this paper, G will always be a discrete countable group. Two systems .G; X; / and .G; Y; / are isomorphic (i.e., measurably conjugate) if and only if there exist conull sets X 0 X;Y 0 Y and a bijective measurable map X 0 Y 0 1 W ! such that is measurable, and .gx/ g .x/ g G, x X 0. D D 8 2 2 A special class of dynamical systems called Bernoulli systems or Bernoulli shifts has played a significant role in the development of the theory as a whole; it was the problem of trying to classify them that motivated Kolmogorov to in- troduce the mean entropy of a dynamical system over ޚ [Kol58], [Kol59]. That is, Kolmogorov defined for every system .ޚ; X; / a number h.ޚ; X; / called the mean entropy of .ޚ; X; / that quantifies, in some sense, how “random” the system is. His definition was modified by Sinai [Sin59]; the latter has become standard. The lectures notes [Roh67] are a classical reference. Modern references include [Pet83], [Rud90] and [Gla03]. Bernoulli shifts also play an important role in this paper, so let us define them. Let .K; / be a standard Borel probability space. For a discrete countable
1387 1388 LEWISPHYLIPBOWEN group G, let KG Q K be the set of all functions x G K with the D g G W ! product Borel structure and2 let G be the product measure on KG. G acts on KG by .gx/.f / x.g 1f / for x KG and g; f G. This action is measure- D 2 2 preserving. The system .G; KG; G/ is the Bernoulli shift over G with base .K; /. It is nontrivial if is not supported on a single point. Before Kolmogorov’s seminal work [Kol58], [Kol59], it was unknown whether all nontrivial Bernoulli shifts over ޚ were measurably conjugate to each other. He proved that h.ޚ;Kޚ; ޚ/ H./ where H./, the entropy of is defined as fol- D lows. If there exists a finite or countably infinite set K K such that .K / 1 0 0 D then X H./ . k / log.. k // D f g f g k K 2 0 where we follow the convention 0 log.0/ 0. Otherwise, H./ . Thus two D D C1 Bernoulli shifts over ޚ with different base measure entropies cannot be measurably conjugate. The converse was proven by D. Ornstein in the groundbreaking papers [Orn70a], [Orn70b]. That is, he proved that if two Bernoulli shifts .ޚ;Kޚ; ޚ/, .ޚ;Lޚ; ޚ/ are such that H./ H./ then they are isomorphic. D In [Kie75], Kieffer proved a generalization of the Shannon-McMillan theorem to actions of a countable amenable group G. In particular, he extended the defini- tion of mean entropy from ޚ-systems to G-systems. This leads to the generalization of Kolmogorov’s theorem to amenable groups. In the landmark paper [OW87], Ornstein and Weiss extended most of the classical entropy theory from ޚ-systems to G-systems where G is any countable amenable group. In particular, they proved that if two Bernoulli shifts .G; KG; G/, .G; LG; G/ over a countably infinite amenable group G are such that H./ D H./ then they are isomorphic. Thus Bernoulli shifts over G are completely clas- sified by base measure entropy. Now let us say that a group G is Ornstein if H./ H./ implies .G; KG; G/ D is isomorphic to .G; LG; G/ whenever .K; / and .L; / are standard Borel prob- ability spaces. By the above, all countably infinite amenable groups are Ornstein. Stepin proved that any countable group that contains an Ornstein subgroup is itself Ornstein [Ste75]. It is unknown whether every countably infinite group is Ornstein. In [OW87], Ornstein and Weiss asked whether all Bernoulli shifts over a non- amenable group are isomorphic. The next result shows that the answer is ‘no’:
THEOREM 1.1. Let G s1; : : : ; sr be the free group of rank r. If .K1; 1/, D h i G G .K2; 2/ are standard probability spaces with H.1/ H.2/< then .G; K1 ; 1 / G G C 1 is measurably conjugate to .G; K ; / if and only if H.1/ H.2/. 2 2 D AMEASURE-CONJUGACYINVARIANTFORFREEGROUPACTIONS 1389
The reason Ornstein and Weiss thought the answer might be ‘yes’ is due to a curious example presented in [OW87]. It pertains to a well-known fundamental property of entropy: it is nonincreasing under factor maps. Let .G; X; / and .G; Y; / be two systems. A map X Y is a factor if and .gx/ W ! D D g .x/ for almost every x X and every g G. If G is amenable then the mean 2 2 entropy of a factor is less than or equal to the mean entropy of the source. This is essentially due to Sinai [Si59]. So if Kn 1; : : : ; n and n is the uniform G GD f g probability measure on Kn then .G; K2 ; 2 /, which has entropy log.2/, cannot G G factor onto .G; K4 ; 4 /, which has entropy log.4/. The argument above fails if G is nonamenable. Indeed, let G a; b be a D h i rank 2 free group. Identify K2 with the group ޚ=2ޚ and K4 with ޚ=2ޚ ޚ=2ޚ. Then .x/.g/ x.g/ x.ga/; x.g/ x.gb/ x KG; g G WD C C 8 2 2 2 G G G G is a factor map from .G; K2 ; 2 / onto .G; K4 ; 4 /. This is Ornstein-Weiss’ ex- ample. It is the main ingredient in the proof of the next theorem, which will appear in a separate paper.
THEOREM 1.2. Let G be any countable group that contains a nonabelian free subgroup. Then every nontrivial Bernoulli shift over G factors onto every Bernoulli shift over G.
To prove Theorem 1.1, the following invariant is introduced. Let .X; / be any probability space on which G s1; : : : ; sr , the rank r free group, acts by measure- D h i preserving transformations. Let ˛ A1;:::;An be a partition of X into finitely D f g many measurable sets. Let B.e; n/ G denote the ball of radius n centered at the identity element with respect to the word metric induced by S s 1; : : : ; s 1 . D f 1˙ r˙ g The join of two partitions ˛; ˇ of X is defined by ˛ ˇ A B A ˛; B ˇ . _ D f \ j 2 2 g Let X H.˛/ .A/ log..A//; WD A ˛ 2 r X F .˛/ .1 2r/H.˛/ H.˛ si ˛/; WD C _ i 1 _ D ˛n g˛; WD g B.e;n/ 2 f .˛/ inf F .˛n/: WD n A partition ˛ is generating if the smallest G-invariant -algebra containing ˛ is the -algebra of all measurable sets (up to sets of measure zero). The main theorem of this paper is: 1390 LEWISPHYLIPBOWEN
THEOREM 1.3. Let G s1; : : : ; sr . Let .G; X; / be a system. If ˛ and ˇ D h i are finite measurable generating partitions of X then f .˛/ f .ˇ/. Hence this D number, denoted f .G; X; /, is a measure-conjugacy invariant. Theorem 5.2 below implies that if K < then f .G; KG; G/ H./. This j j 1 D and Stepin’s theorem proves Theorem 1.1. A simple exercise reveals that if r 1, D then f .G; X; / h.G; X; / is Kolmogorov’s entropy. D Here is a brief outline of the paper. In the next section, standard entropy-theory definitions are presented. In Section 3, an equivalence relation, called combinato- rial equivalence, is introduced on the space of finite partitions of X, where .X; / is a standard probability space on which a countable group G acts. We prove that the combinatorial equivalence class of a finite generating partition is dense in the space of all generating partitions. In Section 4, we introduce an operation on partitions called splitting and show that any two combinatorially equivalent partitions have a common splitting. This culminates in a condition sufficient for a function from the space of partitions to ޒ to induce a measure-conjugacy invariant. In Section 5, this condition is shown to hold for the function F defined above. This proves Theorem 1.3. Then we prove Theorem 5.2 (that f .G; KG; G/ H./ if D K < ) and conclude Theorem 1.1. j j 1 2. Some standard definitions
For the rest of this section, fix a standard probability space .X; /.
Definition 1. A partition ˛ A1;:::;An is a pairwise disjoint collection D f n g of measurable subsets Ai of X such that Ai X. The sets Ai are called the [i 1 D partition elements of ˛. Alternatively, they areD called the atoms of ˛. Unless stated otherwise, all partitions in this paper are finite (i.e., n < ). 1 If ˛ and ˇ are partitions of X then we write ˛ ˇ a.e. if for all A ˛ there D 2 exists B ˇ with .AB/ 0. Let ᏼ denote the set of all a.e.-equivalence classes 2 D of finite partitions of X. By a standard abuse of notation, we will refer to elements of ᏼ as partitions. Definition 2. If ˛; ˇ ᏼ then the join of ˛ and ˇ is the partition ˛ ˇ 2 _ D A B A ˛; B ˇ . f \ j 2 2 g Definition 3. Let Ᏺ be a -algebra contained in the -algebra of all measur- able subsets of X. Given a partition ˛, define the conditional information function I.˛ Ᏺ/ X ޒ by j W ! I.˛ Ᏺ/.x/ log .Ax Ᏺ/.x/ j D j where Ax is the atom of ˛ containing x. Here .Ax Ᏺ/ X ޒ is the condi- j W ! tional expectation of Ax , the characteristic function of Ax, with respect to the -algebra Ᏺ. AMEASURE-CONJUGACYINVARIANTFORFREEGROUPACTIONS 1391
The conditional entropy of ˛ with respect to Ᏺ is defined by Z H.˛ Ᏺ/ I.˛ Ᏺ/.x/ d.x/: j D X j If ˇ is a partition then, by abuse of notation, we can identify ˇ with the - algebra equal to the set of all unions of partition elements of ˇ. Through this identification, I.˛ ˇ/ and H.˛ ˇ/ are well-defined. Let I.˛/ I.˛ ∅;X / and j j D jf g H.˛/ H.˛ ∅;X /. D jf g LEMMA 2.1. For any two partitions ˛; ˇ and for any two -algebras Ᏺ1; Ᏺ2 with Ᏺ1 Ᏺ2, H.˛ ˇ/ H.˛/ H.ˇ ˛/; _ D C j H.˛ Ᏺ2/ H.˛ Ᏺ1/: j Ä j Proof. This is well-known. For example, see [Gla03, Prop. 14.16, p. 255]. Definition 4 (Rokhlin distance). Define d ᏼ ᏼ ޒ by W ! d.˛; ˇ/ H.˛ ˇ/ H.ˇ ˛/ 2H.˛ ˇ/ H.˛/ H.ˇ/: D j C j D _ By [Par69, Th. 5.22, p. 62] this defines a distance function on ᏼ. If G is a group acting by measure-preserving transformations on .X; / then the action of G on ᏼ is isometric. Thus, if g G, ˛; ˇ ᏼ then d.g˛; gˇ/ d.˛; ˇ/. From now on, 2 2 D we consider ᏼ with the topology induced by d. ; /. Definition 5. Let G be a group acting on .X; /. Let ˛ be a partition of X. Let †˛ be the smallest G-invariant -algebra containing ˛. Then ˛ is generating if for any measurable set A X there exists a set A †˛ such that .AA / 0. 0 2 0 D Let ᏼgen ᏼ denote the set of all generating partitions. 3. Combinatorially equivalent partitions
For this section, fix a countable group G and an action of G on a standard probability space .X; / by measure-preserving transformations. F W Definition 6. Given ˛ ᏼ and F G finite, let ˛ f F f ˛. 2 D 2 Definition 7. If ˛; ˇ ᏼ are such that for all A ˛ there exists B ˇ with 2 2 2 .A B/ 0 then we say that ˛ refines ˇ and denote it by ˛ ˇ. Equivalently, D ˇ is a coarsening of ˛. Definition 8. Let ˛; ˇ ᏼ. We say that ˛ is combinatorially equivalent to ˇ 2 L M if there exist finite sets L;M G such that ˛ ˇ and ˇ ˛ . Let ᏼeq.˛/ ᏼ Ä Ä denote the set of partitions combinatorially equivalent to ˛ The goal of this section is to prove Theorem 3.3 below: If ˛ is a generating partition then ᏼeq.˛/ is dense in the subspace of all generating partitions. 1392 LEWISPHYLIPBOWEN
LEMMA 3.1. Let ˛ be a generating partition and ˇ B1;:::;Bm ᏼ. Let D f g 2 " > 0. Then there exists a partition ˇ0 B10 ;:::;Bm0 and a finite set L G such L D f g that ˛ ˇ and for all i 1 : : : m, .Bi B / ". 0 D i0 Ä Proof. Since ˛ is generating, there exists a finite set L G such that for every L i 1; : : : ; m , there is a set Bi00, equal to a finite union of atoms of ˛ , such that 2 f g " .Bi B / < . For i 1 : : : m 1, let i00 m D [ B B B : i0 WD i00 j00 j i ¤ [ [ B X B B B B : m0 WD i0 D m00 [ i00 \ j00 i
L 1 L .Bi / < .Bi Af .i//: pı L Since the atoms Bi are pairwise disjoint, it follows that [ Á 1 [ Á BL < BL A : i p i f .i/ i>r ı i>r Since .BL A / 0, it must be that BL A A A , up to a set i f0 .i/ D i f .i/ f0 .i/ f .i/ of measure zero. So, [ Á 1 [ Á BL A A i Ä p f0 .i/ f .i/ i>r ı i>r npı < ": Ä THEOREM 3.3. If ˛ is a generating partition then
ᏼgen ᏼeq.˛/: In other words, the subspace of partitions combinatorially equivalent to ˛ is dense in the space of all generating partitions.
Proof. Let ˛ A1;:::;An and ˇ B1;:::;Bm ᏼgen. Without loss D f g D f g 2 of generality, we assume that .Ai / > 0 for all i 1 : : : n. Let " > 0. By the D previous lemma, there exists a finite set L G such that the atoms of ˇL L L D B ;:::;B can be ordered so that there exists an r 1; : : : ; mL and a func- f 1 mL g 2 f g tion f 1; 2; : : : r 1; 2; : : : ; n so that for all i 1; : : : ; r , W f g ! f g 2 f g .BL A / i \ f .i/ L 1 " .Bi / and [ LÁ (1) Bi < ": i>r 1394 LEWISPHYLIPBOWEN
By choosing " small enough (if necessary) we may assume that f maps onto 1 1; 2; : : : ; n (for example, by choosing " to be smaller than 2 .Aj / over all f g L L j 1 : : : n). By definition of ˇ , mL mj j. If necessary, we may assume D L L Ä that mL mj j after modifying ˇ by adding to it several copies of the empty set. D L That is, for some i, it may occur that B ∅. i D Let ı > 0 be such that ı < ". By Lemma 3.1 there exists a partition M D C1;:::;Cm and a finite set M G such that ˛ and .Ci Bi / < ı f g Ä for all i. By choosing ı small enough we may assume the following. Let L D C L;:::;C L . Then, after reordering the atoms of L if necessary, f 1 mL g mL [ Á (2) C L BL ": j j Ä j 1 D Let
L C x Ci if x C for some j then x A i0 D f 2 j 2 j 2 f .j /g mL [ L Ci C A : D \ j \ f .j / j 1 D Let Ci;j Ci Aj C . Let D \ i0 1 C i 1 : : : m Ci;j 1 i; j m : D f i0 j D g [ f j Ä Ä g M L LM Note that 1 .˛ / ˛ where LM lm l L; m M . We claim Ä D D f j 2 2 g that 1 is combinatorially equivalent to ˛. Let †1 be the smallest G-invariant collection of subsets of X that is closed under finite intersections and unions and contains the atoms of 1. It suffices to show that every atom of ˛ is in †1. Observe Sm that, for each i, Ci Ci0 j 1 Ci;j . Hence, Ci †1. Therefore the atoms of L D [ D 2 are also in †1. Since f maps onto 1; 2; : : : ; n , the definition of C implies f g i0 L C Ap C C f .j / p : i0 \ D [f i0 \ j j D g So C Ap is in †1 for all i; p. Now Ci Ap Ci;p .C Ap/. So Ci Ap †1 i0 \ \ D [ i0 \ \ 2 for all i; p. Since m [ Ap Ci Ap; D \ i 1 D Ap †1. Since p is arbitrary, this proves the claim. Thus 1 ᏼeq.˛/. 2 2 We claim that .C Ci / 3" for all i. By definition, i0 Ä mL [ L C Ci Ci C C A : i0 D i0 j f .j / j 1 D AMEASURE-CONJUGACYINVARIANTFORFREEGROUPACTIONS 1395
For each j , C L A .C L BL/ .BL A /: j f .j / j j [ j f .j / Thus,
mL r [ L L [ L [ L (3) C Ci .C B / .B A / .B A /: i0 j j [ j f .j / [ j f .j / j 1 j 1 j >r D D If j r, then by definition of r, Ä .BL A / j \ f .j / L 1 ": .Bj / This implies
.BL AL / ".BL/: j f .j / Ä j Thus r [ Á X (4) BL AL ".BL/ ": j f .j / Ä j Ä j 1 j D Equations (3), (2), (4) and (1) imply the claim. Since ı<" and .Ci Bi /<ı for all i, the above claim implies that .C Bi / i0 Ä 4" for all i. Thus we have shown that for every " > 0, there exists a parti- tion 1 C10 ;:::;Cm0 ;::: , combinatorially equivalent to ˛, containing at most 2D f g m m partition elements and such that .C Bi / < 4" for i 1 : : : m. This C i0 D implies that ˇ is in the closure of ᏼeq.˛/. Since ˇ is arbitrary this implies the theorem.
4. Splittings
In this section, G can be any finitely generated group with finite symmetric generating set S. Let .X; / be a standard probability space on which G acts by measure-preserving transformations.
Definition 9. Let ˛ be a partition. A simple splitting of ˛ is a partition of the form ˛ sˇ where s S and ˇ is a coarsening of ˛. D _ 2 A splitting of ˛ is any partition that can be obtained from ˛ by a sequence of simple splittings. In other words, there exist partitions ˛0, ˛1; : : : ; ˛m such that ˛0 ˛, ˛m and ˛i 1 is a simple splitting of ˛i for all 1 i < m. D D C Ä If is a splitting of ˛ then ˛ and are combinatorially equivalent. The splitting concept originated from Williams’ work [Wil73] in symbolic dynamics. 1396 LEWISPHYLIPBOWEN
Definition 10. The Cayley graph of .G; S/ is defined as follows. The vertex set of is G. For every s S and every g G there is a directed edge from g to 2 2 gs labeled s. There are no other edges. The induced subgraph of a subset F G is the largest subgraph of with vertex set F . A subset F G is connected if its induced subgraph in is con- nected.
LEMMA 4.1. If ˛; ˇ ᏼ, ˛ refines ˇ and F G is finite, connected and 2 contains the identity element e then _ ˛ fˇ _ f F 1 2 is a splitting of ˛. Proof. We prove this by induction on F . If F 1 then F e and the j j j j D D f g statement is trivial. Let f0 F e be such that F1 F f0 is connected. To 2 f g D f g see that such an f0 exists, choose a spanning tree for the induced subgraph of F . Let f0 be any leaf of this tree that is not equal to e. W By induction, ˛1 ˛ 1 fˇ is a splitting of ˛. Since F is connected, f F WD _ 2 1 there exists an element f1 F1 and an element s1 S such that f1s1 f0. Since 1 2 2 D f1 F1, ˛1 refines .f ˇ/. Thus 2 1 _ 1 1 1 ˛ fˇ ˛1 f ˇ ˛1 s .f ˇ/ _ D _ 0 D _ 1 1 f F 1 2 is a splitting of ˛.
PROPOSITION 4.2. Let ˛; ˇ be two combinatorially equivalent generating partitions. Then there is an n 0 such that _ ˛n g˛ D g B.e;n/ 2 is a splitting of ˇ. Here B.e; n/ is the ball of radius n centered at the identity element in G with respect to the word metric induced by S. Of course, ˛n is also a splitting of ˛. This proposition is a variation of a result that is well-known in the case G ޚ D in the context of subshifts of finite-type. For example, see [LM95, Th. 7.1.2, p. 218]. It was first proven in [Wil73]. Proof. Let L;M G be finite sets such that ˛ ˇL and ˇ ˛M . Let l; m ގ Ä Ä 2 be such that L B.e; l/ and M B.e; m/. So ˛ ˇl and ˇ ˛m. Since balls Ä Ä are connected and ˛ ˇl , the previous lemma implies ˇl ˛m l is a splitting of Ä _ C ˇl , and therefore, is a splitting of ˇ. But ˇl ˛m l .ˇ ˛m/l ˛m l . _ C D _ D C AMEASURE-CONJUGACYINVARIANTFORFREEGROUPACTIONS 1397
THEOREM 4.3. Let ˆ ᏼ ޒ be any continuous function. Suppose that ˆ is W ! monotone decreasing under splittings; i.e., if is a splitting of ˛ then ˆ./ ˆ.˛/. Ä Define ᏼ ޒ by W ! .˛/ lim ˆ.˛n/ inf ˆ.˛n/: D n D n !1 Then, for any two finite generating partitions ˛1 and ˛2, .˛1/ .˛2/. So D we may define .G; X; / .˛/ for any finite generating partition ˛. The number D .G; X; / is a measure-conjugacy invariant.
Proof. Let ˛ and ˇ be two combinatorially equivalent finite partitions. We claim that .˛/ .ˇ/. To see this, suppose, for a contradiction, that .˛/ < D .ˇ/. Then there exists an n 0 such that ˆ.˛n/ < .ˇ/. But by the previous proposition, there is an m 0 such that ˇm is a splitting of ˛n which implies ˆ.˛n/ ˆ.ˇm/ .ˇ/, a contradiction. So .˛/ .ˇ/. n D For n 0 and ˛ ᏼ, let ˆn.˛/ ˆ.˛ /. Since ˆ is continuous and the n 2 D map ˛ ˛ is also continuous, it follows that ˆn is continuous. Since .˛/ 7! D infn ˆn.˛/, it follows that is upper semi-continuous, i.e., if ˇn is a sequence f g of partitions converging to ˛ then lim sup .ˇn/ .˛/. n Ä Now let ˛; ˇ be two finite generating partitions. By Theorem 3.3, there exist
finite partitions ˇn n1 1 combinatorially equivalent to ˇ such that ˇn n1 1 con- f g D f g D verges to ˛. So .ˇ/ lim sup .ˇn/ .˛/. Similarly, .˛/ .ˇ/. So D n Ä Ä .˛/ .ˇ/. D 5. The f -invariant
In this section, G s1; : : : ; sr . Let .X; / be a standard probability space on D h i which G acts by measure-preserving transformations and let S s 1; : : : ; s 1 . D f 1˙ r˙ g Note S 2r. Let F ᏼ ޒ be defined as in the introduction. j j D W ! PROPOSITION 5.1. Let ˛ ᏼ. If is a splitting of ˛ then F ./ F .˛/. 2 Ä Proof. By induction, it suffices to prove the proposition in the special case in which is a simple splitting of ˛. So let ˛ tˇ for some t S and coarsening D _ 2 ˇ of ˛. For any s S, 2 H. s/ H.˛ s˛/ H. s ˛ s˛/ _ D _ C _ j _ H.˛ s˛/ H.s ˛ s˛/ H. ˛ s˛ s/ D _ C j _ C j _ _ H.˛ s˛/ H. ˛ s 1˛/ H. ˛ s˛/: Ä _ C j _ C j _ The last inequality occurs because is G-invariant, so
H.s ˛ s˛/ H. ˛ s 1˛/: j _ D j _ 1398 LEWISPHYLIPBOWEN
Since H./ H.˛/ H. ˛/, the above implies D C j F ./ .1 2r/ H.˛/ H. ˛/ Ä C j r X H.˛ s˛/ H. ˛ s 1˛/ H. ˛ s˛/ C _ C j _ C j _ i 1 D X F .˛/ .1 2r/H. ˛/ H. ˛ s˛/: D C j C j _ s S 2 Since ˛ t˛, H. ˛ t˛/ 0. Hence Ä _ j _ D X F ./ F .˛/ .1 2r/H. ˛/ H. ˛ s˛/ Ä j C j _ s S t 2 f g X Á H. ˛ s˛/ H. ˛/ 0: D j _ j Ä s S t 2 f g Theorem 1.3 now follows from the proposition above and Theorem 4.3.
Definition 11. Let K be a finite set and a probability measure on K. Let KG be the product space with the product measure G. The system .G; KG; G/ is called the Bernoulli shift over G with base measure . Let A x KG x.e/ k where e denotes the identity element in G. k D f 2 j D g Then ˛ A k K is the Bernoulli partition associated to K. It is generating D f k j 2 g and H./ H.˛/, by definition. D
THEOREM 5.2. Let G s1; : : : ; sr be the free group of rank r. Let K be a D h i finite set and a probability measure on K. Then
f .G; KG; G/ H./: D
Proof. Let ˛ be the Bernoulli partition associated to K. Let g1; : : : ; gn be n distinct elements of G. It follows from the Bernoulli condition that the collection n gi ˛ of partitions is independent. This means that if j 1; : : : ; n K is f gi 1 W f g ! any functionD then n n G \ Á Y G gi A .A /: j.i/ D j.i/ i 1 i 1 D D It is well-known that this implies
n n _ Á X H gi ˛ H.gi ˛/ nH.˛/: D D i 1 i 1 D D AMEASURE-CONJUGACYINVARIANTFORFREEGROUPACTIONS 1399
See, for example, [Gla03, Prop. 14.19, p. 257]. So for any k 1, 1 X Á F .˛k/ H.˛k s˛k/ . S 1/H.˛k/ D 2 _ j j s S 2 1 X Á B.e; k/ B.s; k/ H.˛/ . S 1/ B.e; k/ H.˛/: D 2 j [ j j j j j s S 2 Suppose r > 1. Then, since G s1; : : : ; sr is free, it is a short exercise to D h i compute: . S 1/k 1 B.e; k/ 1 S j j ; j j D C j j S 2 j j . S 1/k 1 1 B.e; k/ B.s; k/ 2 j j C j [ j D S 2 j j for all s S. Thus, 2 . S 1/k 1 1 . S 1/k 1Á F .˛k/ H.˛/ S j j C . S 1/ . S 1/ S j j D j j S 2 j j j j j j S 2 j j j j H.˛/: D If r 1, then B.e; k/ 2k 1 and B.e; k/ B.s; k/ 2k 2. So it follows in D j jk D C j [ j D C k a similar way that F .˛ / H.˛/. Thus f .G; X; / limk F .˛ / H.˛/ D D !1 D D H./.
Proof of Theorem 1.1. By Stepin’s theorem [Ste75], if .K1; 1/, .K2;2/ G G are standard Borel probability spaces with H.1/ H.2/ then .G; K1 ; 1 / is G G D measurably conjugate to .G; K2 ; 2 /. G G Suppose .K1; 1/, .K2; 2/ are Borel probability spaces such that .G; K1 ; 1 / G G is measurably conjugate to .G; K2 ; 2 /. Let .L1; 1/; .L2; 2/ be probability spaces with L1 L2 < and H.i / H.i / for i 1; 2. By Stepin’s j Gj CG j j 1 D G DG theorem, .G; Li ; i / is measurably conjugate to .G; Ki ; i /. By the above G G theorem, f .G; L ; / H.i /. Since f is a measure-conjugacy invariant, i i D H.1/ H.2/. D 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.
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(Received February 14, 2009)
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