AnnalsofMathematics,151(2000),1119{1150 Entropy and mixing for amenable group actions By Daniel J. Rudolph and Benjamin Weiss* Abstract For ¡ a countable amenable group consider those actions of ¡ as measure- preservingtransformationsofastandardprobabilityspace, writtenasfT(cid:176)g(cid:176)2¡ actingon(X;F;„). WesayfT(cid:176)g(cid:176)2¡ hascompletely positive entropy(orsimply cpe for short) if for any flnite and nontrivial partition P of X the entropy h(T;P) is not zero. Our goal is to demonstrate what is well known for actions of Z and even Zd, that actions of completely positive entropy have very strong mixing properties. Let S be a list of flnite subsets of ¡. We say the S spread i i if any particular (cid:176) 6= id belongs to at most flnitely many of the sets S S¡1. i i Theorem 0.1. For fT(cid:176)g(cid:176)2¡ an action of ¡ of completely positive entropy and P any flnite partition, for any sequence of flnite sets S (cid:181) ¡ which spread i we have 1 #Sih((cid:176)2_SiT(cid:176)¡1(P))!i h(P): The proof uses orbit equivalence theory in an essential way and repre- sents the flrst signiflcant application of these methods to classical entropy and mixing. 1. Introduction The goal of this work is to lift a part of the theory of K-actions in the class of measure-preserving transformations of standard probability spaces to actions of countable amenable groups in particular to show that they must be mixing and in fact mixing of all orders. The standard proof of this for transformations (actions of Z in our vocabulary) or more generally for actions of Zd relies on the existence of the past-algebra and tail-algebra of a partition relative to the action. It seems there is no such good notion for actions of general groups, in particular discrete amenable groups. There are two reasons ⁄BothauthorsthanktheInstituteforAdvancedStudiesattheHebrewUniversityofJerusalem forsupportofthiswork. 1120 DANIEL J. RUDOLPH AND BENJAMIN WEISS for working with actions of countable amenable groups. First there is a rather good analogue of the entropy theory of Z actions for them. This is found in [7] and this as well as the general formulation there of a Rokhlin lemma for such actions are the technical background ideas we will use. Second, these actions are characterized in [1] as precisely those which are orbit-equivalent to actions of Z. We use such an orbit equivalence in an essential way to transfer our problem from ¡ to Z. Neither entropy nor any mixing properties are orbit equivalenceinvariants. Onthefaceofitthiswouldseemtoblocktheuseofsuch an orbit equivalence in this area. We avoid this seeming obstacle by working with entropy and mixing properties relative to a sub (cid:190)-algebra and with orbit equivalences that are measurable with respect to this sub (cid:190)-algebra. We show the former are invariants of the latter. This is the flrst signiflcant application of orbit equivalence theory to classical entropy and mixing questions. We point out that J. Kiefier [5] constructed a version of the Shannon- McMillan Theorem for amenable group actions predating the work of [7] by theintroductionofageneralizedtail-fleld. T.WardandQ.Zhangin[9]usethis methodtoobtainaconditionalentropytheoryforsuchactions. Thisapproach doesnotappearabletodirectlygiveTheorem2.3viathisgeneralizedtail-fleld. The work of Ward and Zhang does give essentially all the results we need for conditional entropy. We give an alternative approach to these results anyway building on the methods of [7] rather than those of [5]. In the next section we give a detailed discussion of our results and the proof of Theorem 2.3 except for two results (Theorems 2.6 and 2.11) that require rather extended argument. In Section 3 we lay out the theory of Rokhlin towers for actions of count- able amenable groups, as developed in [7] and the basic geometry of transfer- enceleadingtoTheorem2.11onspreadsetsandadualresultoninvariantsets. These results are the tools that allow us to pull entropy and mixing properties through an orbit equivalence. In Section 4 we demonstrate the basic entropy theory we will need to prove Theorem 2.6. As promised we also include the simple argument that the direct product of a cpe action and a Bernoulli ac- tion is relatively cpe over the Bernoulli coordinate. This will complete all the technicalities for the proof of Theorem 2.3. In the flnal section we discuss a number of natural extensions of our work here. These include generalizing our work here to actions of continuous groups and several questions concerning amenable groups that might be answerable using orbit equivalence methods. 2. Statements of results and proof of main theorem Let ¡ be a countable discrete amenable group. For our purposes the most natural meaning for this is that ¡ possesses a F¿lner sequence of sets H . We i will go into more detail on this later. Su–ce it here to say that these provide AMENABLE CPE-ACTIONS MIX 1121 a good analogue for intervals, squares, or boxes of indices in Zd for use as both averaging sets for ergodic theorems and for index sets for Rokhlin-like lemmas. Suppose T = fT(cid:176)g(cid:176)2¡ is an action of ¡ by measure-preserving transformations on a standard probability space (X;F;„). If P is a flnite partition of X one can deflne the entropy of the process (T;P) as 1 h(T;P) = il!im1 #Hih((cid:176)2_HiT(cid:176)¡1(P)): That this limit exists and is independent of the choice of F¿lner sequence is a part of the standard machinery found in [7]. Deflnition 2.1. We say a ¡-action T has completely positive entropy (or simply is cpe for short) if for any nontrivial flnite partition P, h(T;P) > 0: This notion is analogous to one of the many deflnitions of K-ness of a single measure-preserving transformation. As said earlier we use this more precise vocabulary to avoid confusion among all these deflnitions. Our goal is to demonstrate that if T is cpe then it must be mixing of all orders. We will demonstrate something somewhat stronger, that there is a uniformity in this that provides a condition equivalent to cpe. We now formulate this. In a countable group ¡ we say a sequence of sets S spreads if each (cid:176) 6= id belongs i to at most flnitely many of the sets S S¡1 and for a flnite set K (cid:181) ¡ we say i i S is K-spread if for any (cid:176) 6= (cid:176) 2 S we have (cid:176) (cid:176)¡1 2= K. If ¡ is Z, for a i 2 1 2 sequence of sets to spread simply means the gaps between consecutive terms in the sets grow. The following result is well known for cpe actions of a single transformation and is what we will generalize here. Theorem 2.2. For T a measure-preserving transformation, T is a K-automorphism (is cpe) if and only if for any flnite partition P and " > 0 there is an N so that for any flnite subset S = fs < s < ::: < s g (cid:181) Z with 1 2 t s ¡s > N for all i, i+1 i fl fl fl 1 fl flfl#Sh((cid:176)_2ST(cid:176)¡1(P)¡h(P))flfl < ": In Theorem 2.2 the conclusion can be written in a couple of equivalent forms. First a form that is a priori weaker: 1 h(_j2ST¡j(P)) > h(P)¡" #S and now a form that is a priori stronger: dt(_j2ST¡j(P);Bt(P)) < " 1122 DANIEL J. RUDOLPH AND BENJAMIN WEISS whereB (P)isanindependentandidenticallydistributedsequenceoftrandom t variables each with the distribution of P. It is quite standard to show that these three are all in fact equivalent. The argument is on flnite lists of random variables and hence remains true for actions of countable groups. For actions of a single transformation T one can state another equivalent condition, that the sequence of processes (Tn;P) converge in d to B(P), the independent and identically distributed process with time-zero distribution that of P. Thislastformulationmaynotmakesenseforanarbitrarydiscreteamenable group, as it may not have spread co-flnite subgroups. But if it does we will be able to reach this conclusion as well. Here is the goal of our work: Theorem 2.3. For T a cpe action of the discrete amenable group ¡ and any flnite partition P and " > 0 there is a flnite subset K (cid:181) ¡ so that for any flnite set S that is K-spread fl fl fl 1 fl flfl#Sh(_(cid:176)2ST(cid:176)¡1(P))¡h(P)flfl < ": To demonstrate Theorem 2.3 we \transfer" the problem to actions of Z by the fundamental result of [1], that any discrete amenable group action is orbit-equivalent to an action of Z and by our development here of a relative entropy theory for actions of ¡. Deflnition 2.4. For T a measure-preserving action of ¡, A a T-invariant sub-(cid:190)-algebra and P a flnite partition we deflne the conditional entropy of the process (T;P) given the algebra A to be h(T;PjA) = inffh(T;P _Q)¡h(T;Q) : Q is A-measurableg: This is of course just the standard deflnition for actions of Z. We can also consider orbit equivalences that are A-measurable. Deflnition 2.5. Let T be a measure-preserving and free action of ¡ and A a T-invariant sub-(cid:190)-algebra. Suppose S is a free action of some perhaps 0 difierent group ¡ but having the same orbits as T. We say the orbit change from T to S is A-measurable if for each (cid:176)0 2 ¡0 the functions (cid:176)(x) deflned by T(cid:176)(x)(x) = S(cid:176)0(x) are A-measurable. For example, suppose T is actually a direct product of two free ¡-actions T £T and there is an action U of some ¡0 with the same orbits as T . The 1 2 2 action of U then can be lifted uniquely to an action U^ with the same orbits AMENABLE CPE-ACTIONS MIX 1123 as T1£T2. If U(cid:176)0(x2) = T2;(cid:176)(x2)(x2) then we can (and must) set U^(cid:176)0(x1;x2) = (T £T ) (x ;x ). In this case it is clear that the orbit change from T £T 1 2 (cid:176)(x2) 1 2 1 2 to U^ is measurable with respect to the second coordinate (cid:190)-algebra B . 2 Note also that if the orbit change from T to U is A-measurable then A is a U-invariant sub-(cid:190)-algebra. With these comments in mind we can now state the core technical fact we will use to obtain Theorem 2.3. Theorem 2.6. Suppose T is a free and ergodic action of a countable and discrete amenable group ¡ and A is a T-invariant sub-(cid:190)-algebra. Suppose also 0 that U is a free (and necessarily ergodic) action of ¡ with the same orbits 0 as T (¡ is necessarily amenable). Suppose the orbit change from T to U is A-measurable. Then for any flnite partition P we conclude h(T;PjA) = h(U;PjA): Noticethatthisresultisinsharpcontrasttotheabsolutecase. Allergodic actionsofZareorbit-equivalentsoentropyitselfcannotbeinvariant. Herethat wedealwithconditionalentropyandrequiretheorbitchangetobemeasurable with respect to the sub-(cid:190)-algebra on which we condition changes the situation entirely. Inthissituationtheorbitchangenolongerhasthefreedomtomodify theconditionalentropy,thatistosaytheentropythatT hasaboveandbeyond the entropy on the sub-(cid:190)-algebra. With this result in hand and T some cpe action of ¡ we \transfer" the problem to Z by considering a direct product action T £T2 = fT(cid:176) £T2;(cid:176)g(cid:176)2¡; where all we really need of T is that this direct product should still be ergodic 2 and the action of T should be free and hence orbit-equivalent to an action of 2 Z. As described above this means the orbit equivalence of T to some Z-action 2 U will lift to an ergodic map U^ giving a Z action orbit-equivalent to T £T . 2 Moreover the orbit change from T £T to U^ is measurable with respect to its 2 second coordinate. InthedirectproductofT andT considertherelativePinskeralgebraover 2 the second coordinate, that is to say the (cid:190)-algebra generated by all partitions P withh(T£T ;PjB ) = 0:WhenthisalgebraisjustB theactionissaidtobe 2 2 2 relatively cpe over B . This relative Pinsker algebra contains both B and the 2 2 Pinsker algebra of T. It is a general fact in a direct product like this that the relative Pinsker algebra is precisely the span of these two algebras. (see [2]). When T is a Bernoulli action there is a particularly easy proof so we include 2 this (see Corollary 4.11). In our case this means that the Pinsker algebra of T £ T is just that of T which is to say T £ T is relatively cpe over B . 2 2 2 2 1124 DANIEL J. RUDOLPH AND BENJAMIN WEISS We conclude that U^ must be relatively cpe (relatively K) over the algebra B as well. The theory of relative K-ness for actions of Z is well-developed 2 and parallels completely the nonrelative case. We will quite regularly write expressions of the form E(PjA) where P is a flnite partition and A is a sub-(cid:190)- algebra. Consider P as a map from X to the labels f1;2;:::;ng. By E(PjA) we will mean the probability vector fE(1P¡1(1)jA);:::;E(1P¡1(n)jA)g: ForU anergodicmeasure-preservingtransformation,AaU-invariantsub- (cid:190)-algebra, P a flnite partition, and S a flnite subset of Z we deflne L(_i2SU¡i(P)jA) = H(E(_i2SU¡i(P)jA)) = E(I(_i2SU¡i(P)jA)jA): That is to say, we calculate the conditional probabilities given A of the various sets in _i2SU¡i(P) and then take the entropy of this probability vector. (For our purposes h calculates the entropy of a partition relative to a flxed measure and H calculates the entropy of a probability vector.) The following result, due to M. Rahe [8] is fundamental to obtaining our result. Notice that when A is the trivial algebra it reduces to Theorem 2.2. Theorem2.7(M.Rahe[8]). SupposeU^ isanergodicmeasure-preserving transformation, A is a U^-invariant sub-(cid:190)-algebra, and U^ is relatively K with respect to A. Then for any flnite partition P and " > 0 there is an N so that for any flnite subset S (cid:181) N with js ¡s j > N for all i 6= j, i j (cid:176) (cid:176) (cid:176) 1 1 X (cid:176) (cid:176)(cid:176) L(_i2SU^¡iPjA)¡ L(U^¡i(P)jA)(cid:176)(cid:176) < ": #S #S i2S 1 Notice that in the conclusion all the information functions are A-measur- able but as stated the set S is a constant over the space. We need to consider the possibility that S = S(x) is an A-measurable choice of the set of indices along which the calculation is made. We also will need to loosen up the con- dition that all gaps in the sequence are at least N. Deflnition 2.8. SupposeS(x)isameasurablechoiceofak-elementsubset of Z. We say S(x) is N-quasi-spread if for all x outside a subset of measure less than 1=N, there is a subset S0(x) (cid:181) S(x) with #S0(x)=#S(x) > 1¡1=N and for all distinct s;s0 2 S0(x) we have js¡s0j ‚ N: We can generalize this idea to a countable discrete group ¡ by flxing a listing of its elements (cid:176) ;(cid:176) ;::: = ¡. Suppose S(x) = fs (x);:::;s (x)g is 1 2 1 k a measurable choice of a k-element subset of ¡. We say S(x) is N-quasi- spread if for all x outside a subset of measure less than 1=N, there is a subset AMENABLE CPE-ACTIONS MIX 1125 S0(x) (cid:181) S(x) with #S0(x)=#S(x) > 1¡1=N and for all distinct s;s0 2 S0(x) we have s¡1s0 2= f(cid:176) ;(cid:176) ;:::;(cid:176) g: 1 2 N Although this latter deflnition depends explicitly on how we choose to list the elements of the group it is easy to see that if we flx two listings of the group, to be well quasi-spread for one listing is to be well quasi-spread for the other. WeconjecturethatjustassumingthatS(x)issu–cientlyquasi-spreadwill not be enough to prove Theorem 2.7. Our proof requires us to ask something else of the function that is automatically true for a constant set. Deflnition 2.9. ForS(x)ameasurablechoiceofk-elementsubsetof¡we sayS isuniformrelativetothe¡-actionT iftherearekelementsW = T (x) i si(x) inthefull-groupofT (i.e. theW areone-to-one,ontoandmeasure-preserving) i such that S(x) = fs (x);:::;s (x)g: 1 k Notice that this implies the k image points W (x) are distinct. i The following simple observation is one piece of our transference, saying that the notion of uniformity transfers through a rearranging of the orbit. Lemma 2.10. Suppose T is a free and ergodic action of the countable 0 group ¡ and U is an action of another countable group ¡ with the same orbits as T. Suppose S(x) is a k-element set-valued function of X taking values in ¡ and uniform. Let W (x) be k maps whose set of images form the S(x). Each i W can be written as i W (x) = U (x): i vi(x) Letting V(x) = fv (x);v (x);:::;v (x)g, the set-valued function V(x) takes 1 2 k 0 values in the k-point subsets of ¡ and is also uniform. This piece of the transference is transparent. The other piece will require proof. Theorem 2.11. Suppose T is a free action of the countable group ¡ = f(cid:176) ;(cid:176) ;:::g and U is a free action of ¡0 = f(cid:176)0;(cid:176)0;:::g with the same orbits. 1 2 1 2 Given any N 2 N there is an M 2 N so that for any k and S(x) taking values in the k-point subsets of ¡ that is uniform and M-quasi-spread, the set-valued 0 function V(x), taking values in the k-point subsets of ¡ will be uniform and N-quasi-spread. We postpone the proof of Theorem 2.11 to Section 2. In Theorem 2.7 it is enough to ask that the set S be N-quasi-spread as 0 the indices outside S will contribute only an error on the order of log#P=N 1126 DANIEL J. RUDOLPH AND BENJAMIN WEISS to each term in the difierence. What is not at all clear though is that we can replace a constant set S with a variable S(x). We will see though that under the assumption of uniformity we can. Theorem 2.12. Suppose U is an ergodic measure-preserving transfor- mation, A is a U-invariant sub-(cid:190)-algebra, and U is relatively K with respect to A. Then for any flnite partition P and " > 0 there is an N so that for any A-measurable function S taking values in the k-point subsets of Z that is both N-quasi-spread and uniform we will have (cid:176) (cid:176) (cid:176)1 1 X (cid:176) (cid:176)(cid:176) L(_i2SU¡iPjA)¡ L(U¡i(P)jA)(cid:176)(cid:176) < ": k k i2S 1 Proof. To say that U is relatively K with respect to the (cid:190)-algebra A is to say that any partition P has a trivial relative tail. That is to say, (cid:181) ¶ \ _ U¡i(P)_A = A N i‚N and hence lim E(Pj_i‚N U¡i(P)_A) = E(PjA): N!1 Assume P is an n-set partition BN = fx : jL(Pj_i‚N U¡i(P)_A)(x)¡L(PjA)(x)j < "=(4logn)g: Choose N su–ciently large that „(B ) > 1¡"=(4logn) and that 1=N N < "=(4logn): Now assume S(x), taking values in the k-point subsets of Z, is both N-quasi-spread and uniform. For each x 2 X rewrite S(x) as s (x) > s (x) > 1 2 ¢¢¢ > s (x);s (x);:::;s (x) where the ordering is chosen so that t(x) is t(x) t(x)+1 k maximal for the properties 1) for i < t(x), s (x)¡s (x) > N; and i i+1 2) for i • t(x), Usi(x)(x) 2 BN: Notice that as S(x) is A-measurable as is B , we can assume that both s (x) N i and t(x) are as well. As S is uniform we can calculate Z #fs (x) : Usi(x)(x) 2 B gd„ ‚ k(1¡"=(4logn)) i N and letting S0(x) (cid:181) S(x) be a maximal set of indices pairwise more than N apart, as S is N-quasi-spread Z #S0(x)d„ ‚ k(1¡"=(2logn)): AMENABLE CPE-ACTIONS MIX 1127 We conclude that Z t(x)d„ ‚ k(1¡3"=(4logn)): We now calculate L(_i2S(x)U¡i(P)jA) one term at a time in the form Xk L(Usi(x)(P)j_ Usj(x)(P)_A): j<i i=1 We immediately conclude that Xk L(_i2S(x)U¡i(P)jA) • L(U¡si(x)(P)jA) i=1 and so we only need an inequality in the other direction. The following inequalities are true almost surely in x. Xk t L(Usi(x)(P)j_ U¡sj(x)(P)_A) j<i i=1 tX(x) ‚ L(Usi(x)(P)j_ U¡j(P)_A) j>si(x)+N i=1 tX(x) ‚ (L(Usi(x)(P)jA)¡"=(4logn)) i=1 Xk ‚ (L(Usi(x)(P)jA)¡"=(4logn))¡(k¡t(x))lnn: i=1 We conclude that Z fl fl fl1 1 X fl flfl L(_i2S(x)U¡i(P)jA)¡ L(U¡i(P)jA)flfld„ k k Zi2S • "=(4logn)+k¡ td„ • ": Combining Theorems 2.12 and 2.11 we reach the following conclusion. Theorem 2.13. Suppose T is a free and ergodic action of a countable amenable group ¡ = f(cid:176) ;(cid:176) ;:::g, A is a T-invariant sub-(cid:190)-algebra restricted to 1 2 which the action of T is still free and relative to which the action of T is cpe, and lastly P is a flnite partition. Given any " > 0 there is an M so that for any k and any A-measurable function S(x) taking values in the k-point subsets of ¡, if S is both uniform and M-quasi-spread then we will have (cid:176) (cid:176) (cid:176)1 1 X (cid:176) (cid:176)(cid:176)kL(_(cid:176)2ST(cid:176)¡1PjA)¡ k L(T(cid:176)¡1(P)jA)(cid:176)(cid:176) < ": (cid:176)2S 1 1128 DANIEL J. RUDOLPH AND BENJAMIN WEISS Proof. Construct an action U of Z that has the same orbits as T and for which the orbit rearrangement is A-measurable. For any S(x) we now obtain a V(x) taking values in the k-point subsets of Z that must be uniform. Notice that L(_(cid:176)2S(x)T(cid:176)¡1(P)jA)(x) = L(_i2V(x)U¡i(P)jA)(x) and for almost every (a.e.) x, for each (cid:176) 2 S(x) there is a unique i 2 V(x) with T (x) = Ui(x) and so (cid:176) L(T(cid:176)¡1(P)jA) = L(U¡i(P)jA): Thus we know (cid:176) (cid:176) (cid:176)1 1 X (cid:176) (cid:176)(cid:176)kL(_(cid:176)2ST(cid:176)¡1(P)jA)¡ k L(T(cid:176)¡1(P)jA)(cid:176)(cid:176) (cid:176) (cid:176)2S 1 (cid:176) (cid:176)1 1 X (cid:176) = (cid:176)(cid:176) L(_i2VU¡i(P)jA)¡ L(U¡i(P)jA)(cid:176)(cid:176) : k k i2V 1 From Theorem 2.11 we know that for any N if M is large enough then V will be N-quasi-spread and Theorem 2.12 completes the theorem. In the case we are interested in T is in fact T £T , the algebra A is the 1 2 second coordinate algebra, P is measurable with respect to the flrst coordinate algebra and S is a constant choice of set. In this case notice that L(_(cid:176)2ST(cid:176)¡1(P)jA) = h(_(cid:176)2ST(cid:176)¡1(P)) and L(T(cid:176)¡1(P)jA) = h(P): From this the proof of Theorem 2.3 is complete. 3. Discrete amenable groups and spread sets In this section we lay out the basic ergodic theory of measure-preserving actions of countable and discrete amenable groups. This material can all be found in [7] or [4]. Our goal is to describe some particular structures that are invariants of orbit equivalence. As described in the introduction we will be \transferring" our work from an action of ¡ to one of Z with the same orbits. We here establish a vocabulary for transferring certain calculations through this transference. The most important of these will be conditional entropy. The transference of these calculations will be made by showing that they can be made on towers. Our flrst step is to lay out the structure of Rokhlin towers as developed in [7]. These build from notions of invariance of sets in ¡.