Entropy sequences and maximal entropy sets

INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 19 (2006) 53–74

NONLINEARITY doi:10.1088/0951-7715/19/1/004

Entropy sequences and maximal entropy sets Dou Dou, Xiangdong Ye and Guohua Zhang Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China E-mail: [email protected], [email protected] and [email protected]

Received 3 May 2005, in final form 31 August 2005 Published 18 October 2005 Online at stacks.iop.org/Non/19/53 Recommended by M Tsuji Abstract The notions of entropy sequences and entropy sets are introduced both in topological and measure-theoretical situations. It turns out that a subset is an entropy set if and only if each n-tuple (not on the diagonal) from the set is an entropy n-tuple and that each maximal entropy set is closed. The systems with only one maximal entropy set are characterized. Moreover, it is proved that if a topological system has positive entropy, then there is a maximal entropy set with uncountably many points, and the topological entropy is the supremum of the entropies over all maximal entropy sets. An example with a maximal entropy set containing only two points is given. Mathematics Subject Classification: 37A05, 37A35

1. Introduction Ergodic theory and topological dynamics exhibit a remarkable parallelism. For example, we speak about measure-theoretical entropy in ergodic theory and topological entropy in topological dynamics. A Kolmogorov system in ergodic theory is an important class and it completely differs from zero entropy systems. To get a topological analogy, Blanchard [B1] introduced the notions of c.p.e. and u.p.e. in topological dynamics. He then naturally defined the notion of entropy pairs and used it to show that a u.p.e. system is disjoint from all minimal zero entropy systems [B2]. Later on, in [B-R] the authors were able to define entropy pairs for an invariant measure and showed that for each invariant measure the set of entropy pairs for this measure is contained in the set of entropy pairs. Blanchard, Glasner and Host [BGH] proved that the converse of [B-R] is also valid, that is, there is an invariant measure such that the set of entropy pairs is contained in the set of entropy pairs for this measure. A characterization of the set of entropy pairs for an invariant measure as the support of some measure was obtained by Glasner [G1]. 0951-7715/06/010053+22$30.00 © 2006 IOP Publishing Ltd and London Mathematical Society Printed in the UK

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To obtain a better understanding of the topological version of a Kolmogorov system, Huang and Ye [HY] introduced the notions of entropy n-tuples (n
2) both in topological and measure-theoretical settings. A local variational relation was proved and the relation between the set of topological entropy n-tuples and the set of entropy n-tuples for an invariant measure was established. Moreover, it was shown that the dynamical behaviours in any neighbourhood of an n-tuple are complicated for a positive entropy system. Now a natural question is: is there some infinite subset such that the dynamical behaviours in any neighbourhood of the set are complicated in a positive entropy system? If there is, how ‘big’ could the subset be? We will answer these questions in this paper. Namely, we introduce the notions of entropy sequences and entropy sets both in topological and measure-theoretical situations. We show that a set is an entropy set if and only if each n-tuple (not on the diagonal) from the set is an entropy n-tuple and that each maximal entropy set is closed. A system with only one maximal entropy set is characterized, i.e. under the assumption the system looks like a topological K system when restricting to the unique maximal entropy set. Moreover, if a topological system has positive entropy, then there is a maximal entropy set with uncountably many points, and the topological entropy is the supremum of the entropies over all maximal entropy sets. We construct an example to show that a maximal entropy set can contain only two points. Even if dynamical entropy is not something one can measure physically, positive entropy is often considered to indicate strong chaoticity features in a dynamical system modelling some phenomena. For example, the authors in [BGKM] showed that positive entropy implies chaos in the sense of Li and Yorke, and in [HY] the authors gave a characterization of positive entropy which implies that in a positive entropy system, the map acts as a Bernoulli system along a positive density subsets of N with respect to two closed subsets of the space. Thus, the more one knows about the related notions like entropy sets, the more one learns about the mathematical theory of chaos. The paper is organized as follows. In section 2, first we introduce background and basic definitions in ergodic theory and topological dynamics, then we introduce the notions of entropy sequences and entropy sets in both situations. Basic properties of an entropy set and a maximal entropy set are discussed. Section 3 is devoted to discussing the existence of entropy sequences in topological systems with positive entropy. Then we focus in section 4 on the properties of maximal entropy sets. In section 5 systems with only one maximal entropy set are characterized. In the last section we construct an example to show that a maximal entropy set can contain only two points.

2. Preliminaries A topological dynamical system (TDS) is a pair (X, T ), where X is a compact metric space and T is a homeomorphism of X onto itself. We say TDS (X, T ) is transitive if for each pair (U, V ) of non-empty open subsets there exists n ∈ N such that U ∩ T −n (V ) = ∅. x0 ∈ X is called a transitive point of (X, T ) if {x0 , T (x0 ), T 2 (x0 ), . . .} is dense in X. Let (X, B, µ, T ) be a measure-theoretic dynamical system (MDS), that is, X is a set, B is a σ -algebra on X, T is a transformation of X and µ is a probability measure which is invariant under T . Let Pµ be the Pinsker σ -algebra. For a finite measurable partition
+∞ −i P, let P− = i=1 T P and Hµ (P|A) be the conditional entropy of P with respect to a given σ -algebra A. As usual, P ∨ T −1 P ∨ · · · ∨ T −(n−1) P is denoted by P0n−1 (T ) or simply P0n−1 .

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For a given TDS (X, T ) and an open cover U of X one defines the topological entropy of U by the usual formula 1 H (U ∨ T −1 (U ) ∨ · · · ∨ T −(n−1) (U )); n the limit exists because of the subadditivity of H (U ) = log inf #(U  ), where the infimum is taken over all subcovers U  of U and # denotes the cardinality. By U  V , we mean that cover U is coarser than the cover V and the same notation will be used for partitions as well. When U  V , we have htop (T , U )  htop (T , V ). The entropy of the TDS (X, T ) is htop (T , U ) = lim

n→+∞

htop (T ) = sup htop (T , U ), where the supremum is taken over all open covers U . Denote by M(X, T ) and Me (X, T ) respectively the set of all invariant probability measures and all ergodic invariant probability measures on (X, BX , T ), where BX is the Borel σ -algebra on X. For a given µ ∈ M(X, T ), the entropy of µ with respect to a finite measurable partition P is defined by hµ (T , P) = lim

n→+∞

1 − − Hµ (Pn−1 0 ) = Hµ (P|P ) = Hµ (P|P ∨ Pµ ). n

The entropy of µ is hµ (T ) = sup hµ (T , P), where P ranges over all finite measurable partitions of (X, BX , T ). The variational principle tells us that htop (T ) =

sup µ∈Me (X,T )

hµ (T ).

The notion of an entropy pair introduced in [B2, B-R] is generalized to an entropy tuple in [HY] both in TDS and MDS. Let E ⊂ X and #E
2. Let U = {U1 , U2 , . . . , Un } be a finite Borel cover of X. We call U an admissible cover with respect to E if for each Ui (1  i  n) there exists xi ∈ E (1  i  n) such that xi is not in the closure of Ui . Similarly one has the definition of an admissible partition if the cover U is replaced by a Borel partition P = {P1 , P2 , . . . , Pn }. Definition 2.1. Let (X, T ) be a TDS and X(n) = X × · · · × X (n times) with n
2. (1) An n-tuple (xi )n1 ∈ X(n) is a topological entropy n-tuple (n-topo) if at least two of the points {xi }ni=1 are different and if whenever Uj are closed mutually disjoint neighbourhoods of distinct points xj , the open cover U = {Ujc : 1  j  n} has positive topological entropy, i.e. htop (T , U ) > 0. (2) (X, T ) has uniform positive entropy of order n (u.p.e. of order n), if every point (xi )n1 ∈ X(n) , not on the diagonal n (X) = {(x)n1 : x ∈ X}, is an n-topo. (3) (X, T ) has u.p.e. of all orders or topological K if it has u.p.e. of order n for every n
2. Notice that an n-tuple (xi )n1 ∈ X (n) is a topological entropy n-tuple if and only if at least two of the points {xi }ni=1 are different and for any admissible finite open cover U with respect to {xi }n1 one has htop (T , U ) > 0. The entropy n-tuple for an invariant measure is defined as the following. Definition 2.2. Let (X, T ) be a TDS and µ ∈ M(X, T ). An n-tuple (xi )n1 is an entropy n-tuple for µ, if for any admissible partition P with respect to {xi }n1 we have hµ (T , P) > 0.

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Let (X, T ) be a TDS. For each n ∈ N : n
2, let En (X, T ) be the set of topological µ entropy n-tuples and En (X, T ) be the set of entropy n-tuples with respect to an invariant measure µ. Recall that for n
2, n (X) = {(x1 , . . . , xn ) ∈ X(n) : x 1 = · · · = xn }. Put ∞ (X) = {(x1 , x2 , . . .) ∈ X(∞) : x1 = x2 = · · ·}, where X (∞) = ∞ i=1 X. n (X) and ∞ (X) are denoted by n and ∞ respectively if there is no confusion. The following propositions are the basic properties of entropy tuples [B2, HY]. Proposition 2.3. Let (X, T ) be a TDS. (1) If U = {U1 , . . . , Un } is an open cover of X with htop (T , U ) > 0, then there are n points xi ∈ Uic for 1  i  n such that (xi )n1 is an n-topo. (2) En (X, T ) is a non-empty closed T (n) -invariant subset of X (n) containing only n-topo and points of n (X). (3) Let π : (Y, S) → (X, T ) be a factor map between TDSs. (1) If (xi )n1 ∈ En (X, T ), then there exist yi ∈ Y , 1  i  n, such that π(yi ) = xi and (yi )n1 ∈ En (Y, S). (2) Conversely if (yi )n1 ∈ En (Y, S) and (π(yi ))n1 ∈ n (X), then (π(yi ))n1 belongs to En (X, T ). (4) Suppose W is a closed T -invariant subset of (X, T ). Then if (xi )n1 is an n-topo of (W, T |W ), it is also an n-topo of (X, T ). We remark that for entropy tuples with respect to an invariant measure, the same proposition holds. Moreover, we have the following propositions [HY]. Proposition 2.4. Let (X, T ) be a TDS. Then µ

(1) En (X, T ) ⊂ En (X, T ) for each µ ∈ M(X, T ) and n
2. µ (2) There is µ ∈ M(X, T ) with En (X, T ) = En (X, T ) for each n
2.  Proposition 2.5. Let (X, T ) be a TDS and µ ∈ M(X, T ). Say µ =  µω dη(ω) is the ergodic decomposition of µ. Then we have µ

µ

(1) En ω (X, T ) ⊆ En (X, T ) for each n
2 and η-almost every ω ∈ .  µ µ (2) If (xi )n1 ∈ En (X, T ), then η({ω ∈ | ni=1 Vi ∩ En ω (X, T ) = ∅}) > 0 for any neighbourhood Vi of xi (1  i  n). Thus for an appropriate choice of  we can require cl(∪{Enµω (X, T ) : ω ∈ })\n = Enµ (X, T ). Entropy sequences and entropy sets are defined as follows. Definition 2.6. Let (X, T ) be a TDS and µ ∈ M(X, T ). (1) A sequence (xi )∞ 1 is an entropy sequence (resp. an entropy sequence for µ), if at least two of the points {xi }∞ i=1 are different and if for any admissible finite open cover U (resp. admissible finite partition P) with respect to {xi }∞ 1 we have htop (T , U ) > 0 (resp. hµ (T , P) > 0). (2) A subset K of X is an entropy set (resp. an entropy set for µ) if #(K)
2 and if for any admissible finite open cover U (resp. admissible finite partition P) with respect to K, one has htop (T , U ) > 0 (resp. hµ (T , P) > 0). An entropy set can be characterized as follows.

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Theorem 2.7. Let (X, T ) be a TDS. (1) A subset K of X is an entropy set if and only if #(K)
2 and for each n
2 and any distinct n-points {x1 , . . . , xn } ⊂ K, the tuple (x1 , . . . , xn ) is an entropy tuple. (2) Let µ ∈ M(X, T ). A subset K of X is an entropy set for µ if and only if #(K)
2 and for each n
2 and any distinct n-points {x1 , . . . , xn } ⊂ K, the tuple (x1 , . . . , xn ) is an entropy tuple for µ. Proof. We show 1 now and the proof of 2 is similar. If K ⊂ X is an entropy set, then for each n
2 and any distinct n-points {x1 , . . . , xn } ⊂ K, any admissible finite open cover U with respect to {x1 , . . . , xn } is also an admissible finite open cover with respect to K. Hence htop (U , T ) > 0 and the tuple (x1 , . . . , xn ) is an entropy tuple. On the other hand, for each n
2, let U = {U1 , U2 , . . . , Un } be any admissible finite open / cl(Ui ), 1  i  n. Notice cover with respect to K. Then there exist xi ∈ X such that xi ∈ that U is an admissible finite open cover with respect to {x1 , . . . , xn }, we have htop (U , T ) > 0. So K is an entropy set.  Remark 2.8. For any given µ ∈ M(X, T ), obviously any entropy set for µ is an entropy set. µ Conversely, choose µ0 ∈ M(X, T ) such that En (X, T ) = En 0 (X, T ) for each n
2 (see proposition 2.4), then any entropy set is also an entropy set for µ0 . It turns out that the closure of an entropy set is again an entropy set. Theorem 2.9. Let (X, T ) be a TDS. (1) If K is an entropy set, then cl(K) is also an entropy set. (2) Let µ ∈ M(X, T ). If K is an entropy set for µ, then cl(K) is also an entropy set for µ. Proof. Let n
2 and {x1 , . . . , xn } ⊂ cl(K) with xi = xj if i = j . Assume for each 1  j  n that xjk → xj when k → ∞ and xjk ∈ K. Then by theorem 2.7, (x1k , . . . , xnk ) ∈ En (X, T ) when k is sufficiently large. As En (X, T ) ∪ n is a closed set, we have (x1 , . . . , xn ) ∈ En (X, T ). This implies that cl(K) is an entropy set. For a given µ ∈ M(X, T ), the same discussion works.  The next result concerns the way how entropy sets are preserved under projection and lift-up. Theorem 2.10. Let π : (X, T ) → (Y, S) be a factor map between TDSs. Then the image of an entropy set under π is an entropy set if it does not collapse to a single point. Conversely, if K ⊂ Y is an entropy set, then there is an entropy set K  ⊂ X with π(K  ) = K. The same is true for an entropy set for an invariant measure µ. Proof. It is easy to see that the image of an entropy set under π is an entropy set if it does not collapse to a single point. Conversely, let K ⊂ Y be an entropy set. Without loss of generality assume that K is n n n closed. Let {yi }∞ 1 be a dense subset of K. For each n
2 let (x1 , x2 , . . . , xn ) be an entropy n tuple such that π(xj ) = yj , 1  j  n. Define inductively, x1 is a limit point of {x1n } with x1ni → x1 , x2 is a limit point of {x2ni } ni ni with x2 j → x2 , x3 is a limit point of {x3 j } and so on. Let K  = {xi : i ∈ N}. It is easy to verify that K  is an entropy set with π(K  ) = K. The same discussion works for an entropy set for an invariant measure. 

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Let (X, T ) be a TDS. If htop (T ) > 0, then En (X, T ) = ∅ for each n
2 (see [B2, HY]), and so there exists an entropy set. By Zorn’s lemma, there are maximal entropy sets in the sense of set inclusion. As an immediate consequence of theorem 2.9 we have the following. Corollary 2.11. Each maximal entropy set is closed. We remark that unlike the minimal subsystems two maximal entropy sets of a TDS can intersect with an arbitrarily given cardinality. For example, let (, σ ) be the full two-sided shift on 2-symbols and A be a closed invariant subset of  with given finite (
2) elements or infinite elements. Take two copies of , namely Xi =  × {i} with i = 1, 2, let X = X1 ∪ X2 and define naturally a map T on X. It is clear that Xi is a maximal entropy set of (X, T ) for i = 1, 2. Identifying (a, 1) and (a, 2) for a ∈ A we get an equivalence relation R and a quotient space X = X/R. Then the intersection of the two maximal entropy sets X1 /R and X2 /R is A/R. To finish this section we state some basic properties of the Pinsker factor which we need in the following sections. We say (X, B, µ, T ) is a Lebesgue system if (X, B, µ) is a Lebesgue space and T : X → X is an invertible measure-preserving transformation. Note that we require that B is complete under µ, i.e. if A ∈ B with µ(A) = 0 then for any C ⊂ A one has C ∈ B. Let (X, B, µ, T ) be a Lebesgue system, Pµ be its Pinsker σ -algebra and π : (X,  B, µ, T ) → (Y, D, ν, S) be the measure-theoretical Pinsker factor of (X, B, µ, T ). Let µ = Y µy dν(y) be the disintegration of µ over ν (see theorem 5.8 of [F]). It is known that for ν-almost every y ∈ Y , µy (π −1 (y)) = 1. µ Recall that for a Lebesgue system (X, B, µ, T ) and each n ∈ N : n
2, λn is an invariant (n) (n) (n) probability measure on (X , B , T ) with  n   n  n      µ (n) λn Ai = E(1Ai |Pµ )(x)dµ(x) = µy Ai dν(y) X i=1

i=1

Y

i=1

for any Ai ∈ B, i = 1, 2, . . . , n, where B(n) = B × · · · × B (n times) and T (n) = T × · · · × T (n times). Note that for a Lebesgue system (X, B, µ, T ), E(1A |Pµ )(x) = µπ(x) (A) for A ∈ B and µ-almost every x ∈ X. Moreover, we have  λµn = µ ×Y · · · ×Y µ (n times) = µ(n) (2.1) y dν(y), Y µ

that is, the disintegration of λn over ν is µ(n) y = µy × · · · × µy (n times). The following result will be used in the following sections, see [G1, HY]. Proposition 2.12. Let (X, T ) be a TDS with µ ∈ M(X, T ). Then supp(λµn )\n = Enµ (X, T ). 3. Entropy sequences Let (X, T ) be a TDS. We know from [HY] that if (X, T ) has positive entropy then for each n
2 there exists an entropy set with n points. Is there an entropy set with infinitely many points? To answer this and related questions, let us first make some preparations. Lemma  3.1. Let (X, T ) be a TDS and µ ∈ M(X, T ) with hµ (T ) > 0. Suppose that µ = Y µy dν(y) is the disintegration of µ over the Pinsker factor (Y, D, ν, S) of (X, Bµ , µ, T ), where (X, Bµ , µ, T ) is the completion of (X, BX , µ, T ). Then for each n ∈ N : n
2 we have  µ µ (1) λn = Y µ(n) y dν(y), and so λn (n ) = 0 if and only if µy is atomless for ν-almost every y ∈ Y.

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µ

(2) If µ ∈ Me (X, T ), then λn is ergodic and µy is atomless for ν-almost every y ∈ Y . µ µ Moreover, λn (n ) = 0, and so supp(λn )\n = ∅. (3) hλµn (T (n) ) = nhµ (T ). Proof. We just consider the case of n = 2, and the same discussion works for other cases.  µ (1) λ2 = Y µy × µy dν(y) follows directly from (2.1). First assume that µy is atomless ν-almost every y ∈ Y . Then by Fubini’s theorem we know, for ν-almost every y ∈ Y   µ µy × µy (2 ) = µy ({x})dµy (x) = 0 ⇒ λ2 (2 ) = µy × µy (2 )dν(y) = 0. X

Y

µ λ2 (2 )

µ λ2 (2 )

Conversely, assume that = 0. That is, Then for ν-almost every y ∈ Y we have  µy × µy (2 ) = µy ({x})dµy (x) = 0.

=

 Y

µy × µy (2 )dν(y) = 0.

X

Thus µy is atomless ν-almost every y ∈ Y . (2) Note that the Pinsker factor is the maximal factor with zero entropy, the entropy of any compact extension of a zero entropy system is zero, and X → Y is a weakly mixing extension. µ Since µ ∈ Me (X, T ), ν is ergodic and so is λ2 . On the other hand, Rohlin’s theorem tells us, either µy is atomless ν-almost every y ∈ Y , or there is some n ∈ N such that for ν-almost every y ∈ Y , µy is an equi-distributed measure on n points. But the latter condition implies that hµ (T ) = 0. Thus µy is atomless ν-almost every y ∈ Y . (see lemma 4.1 of [Z] for details). µ µ Moreover, by part 1 one has λ2 (2 ) = 0, and so supp(λ = ∅.  2 )\2 n−1 (3) First we show that hµ (T , P) = limn→∞ (1/n) Y Hµy (P0 )dν(y) for any finite partition P of (X, Bµ , µ, T ). In fact 1 Hµ (Pn−1 0 |Pµ ) n  1  = lim −E(1A |Pµ )(x) log E(1A |Pµ )(x)dµ(x) n→∞ n n−1 X A∈P0  1  −µy (A) log µy (A)dν(y) = lim n→∞ n Y A∈Pn−1 0  1 Hµy (Pn−1 = lim 0 )dν(y). n→∞ n Y

hµ (T , P) = lim

n→∞

(3.1)

Thus for any finite partitions P1 , P2 of (X, Bµ , µ, T ), say Pi = {Ai1 , . . . , Ail(i) } (i = 1, 2), µ P1 × P2 forms a finite partition of (X (2) , Bµ(2) , λ2 , T (2) ), where P1 × P2 = {A1i × A2j : 1  i  l(1), 1  j  l(2)}. Moreover,  1 hλµ2 (T (2) , P1 × P2 ) = lim Hµy ×µy ((P1 × P2 )n−1 0 )dν(y) (by (3.1)) n→∞ n Y  1 n−1 = lim (Hµy ((P1 )n−1 0 ) + Hµy ((P2 )0 ))dν(y) n→∞ n Y = hµ (T , P1 ) + hµ (T , P2 ). This implies hλµ2 (T (2) ) = 2hµ (T ) (see [W], P99 ).

(3.2) 

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For a Lebesgue system (X, B, µ, T ), let (Y, D, ν, S) be the Pinsker factor of (X, B, µ, T ). µ Define a probability measure λ∞ on (X (∞) , B(∞) ) such that if n ∞   B = [Bn+1 , . . . , Bn+m ] = X × Bn+1 × · · · × Bn+m × X 1

n+m+1

with Bi ∈ B (n + 1  i  n + m) then  n+m  λµ∞ (B) = E(1Bi |Pµ )(x)dµ(x).

(3.3)

X i=n+1

µ

In fact, λ∞ is an invariant measure on (X (∞) , B(∞) , T (∞) ). µ The next theorem tells us some properties of λ∞ . e Theorem 3.2.  Let (X, T ) be a TDS and µ ∈ M (X, T ) with hµ (T ) > 0. Suppose that µ = Y µy dν(y) is the disintegration of µ over the Pinsker factor (Y, Dν , ν, S) of (X, Bµ , µ, T ). Then  µ µ µ (1) λ∞ = Y µ(∞) y dν(y), λ∞ is ergodic and λ∞ (∞ ) = 0. (2) If (x1 , x2 , . . .) ∈ TranT (∞) , then xi = xj when i = j . (3) hλµ∞ (T (∞) ) = ∞.

 µ µ µ Proof. (1) Let λ∞ = Y (λ∞ )y dν(y) be the disintegration of λ∞ over ν. By lemma 3.1 and (3.3), it is clear that for ν-almost every y ∈ Y and any k
2, (λµ∞ )y = µ(k) y × λ(k, y) ∞ for some λ(k, y) on k+1 X. Then for ν-almost every y ∈ Y and the set B of form µ µ (∞) [Bn+1 , . . . , Bn+m ], (λ∞ )y (B) = µ(∞) ν-almost every y ∈ Y , i.e. y (B). Thus (λ∞ )y = µy  µ (∞) λ∞ = Y µy dν(y). µ To show that λ∞ is ergodic, we need to show that X ×Y X ×Y · · · → Y is a weakly mixing extension. This fact can be deduced by using the same idea of proposition 6.3 of [F]. Then µ the ergodicity of λ∞ follows from the ergodicity of µ and the fact. Moreover, since µ ∈ Me (X, T ) and hµ (T ) > 0, it follows from lemma 3.1 that µy is atomless for ν-almost every y ∈ Y . Using Fubini’s theorem we have

   ∞  µ (∞) λ∞ (∞ ) = µy (∞ )dν(y) = µy ({x2 })d µy ((x2 , x2 , . . .)) dν(y) = 0, Y

∞ 2

∞

Y

∞ 2

2

where = {(x2 , x3 , . . .) ∈ 2 X : x2 = x3 = · · ·}. µ µ (2) As (X(∞) , Bµ(∞) , λ∞ , T (∞) ) is ergodic, (supp(λ∞ ), T (∞) ) is transitive. Since ∞ (∞) (∞) is closed T -invariant in X , each (x1 , x2 , . . .) ∈ TranT (∞) is not in ∞ by the fact µ that λ∞ (∞ ) = 0. That is, there are xk = xl for some k < l. Note that for any µ permutation φ : N → N, (xφ(1) , xφ(2) , . . .) ∈ supp(λ∞ ). For any given i = j let φij be a permutation sending i, j to k, l respectively. Since there is a sequence pijn → ∞ with n T pij (x1 , x2 , . . .) → (xφij (1) , xφij (2) , . . .), we have xi = xj . µ (3) Observing that for each n
2, (X(∞) , Bµ(∞) , λ∞ , T (∞) ) is an extension of µ (n) (n) (∞) (n) (X , Bµ , λn , T ), we have hλµ∞ (T ) = ∞ by lemma 3.1 (3).  Let k ∈ N : k
2 and (X, T ) be a TDS. Define q1k : X(∞) → X (k) , (x1 , x2 , . . .) −→ (x1 , . . . , xk ). µ

µ

When µ ∈ M(X, T ), it is easy to obtain q1k (supp(λ∞ )) = supp(λk ), and (x1 , . . . , xk , xk , . . .) µ µ ∈ supp(λ∞ ) if (x1 , . . . , xk ) ∈ supp(λk ). Thus we have the following.

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Corollary 3.3. Let (X, T ) be a TDS with positive entropy. Then there is a maximal entropy set with infinitely many points. Proof. There is some µ ∈ Me (X, T ) with hµ (T ) > 0 by the variational principle. Let µ (x1 , x2 , . . .) be a transitive point of the system (supp(λ∞ ), T (∞) ). Then xi = xj if i = j . We claim {xi : i ∈ N} is an entropy set for µ (and so is an entropy set). In fact, for each n
2, µ µ from q1n (supp(λ∞ )) = supp(λn ) it is easy to see that (x1 , . . . , xn ) ∈ supp(λµn )\n = Enµ (X, T ) (by proposition 2.12),

(3.4)

i.e. (x1 , . . . , xn ) is an entropy tuple for µ. Thus there must exist a maximal entropy set with infinitely many points.  µ

Let (X, T ) be a TDS and µ ∈ M(X, T ). Denote by E∞ (X, T ) and E∞ (X, T ) the set of entropy sequences and µ-entropy sequences respectively. Then we have the following. Theorem 3.4. Let (X, T ) be a TDS and µ ∈ M(X, T ). Then µ supp(λµ∞ )\∞ = E∞ (X, T ), µ

and E∞ (X, T ) is the union of E∞ (X, T ) over all µ ∈ M(X, T ). Moreover, there exists some µ µ0 ∈ M(X, T ) such that E∞ (X, T ) = E∞0 (X, T ). µ

Proof. Let (x1 , x2 , . . .) ∈ supp(λ∞ )\∞ . It makes no difference to assume x1 = x2 . Thus for each n
2, following from (3.4) (x1 , . . . , xn ) is an entropy tuple for µ. Applying theorem µ 2.7 we have (x1 , x2 , . . .) ∈ E∞ (X, T ). µ Now let (x1 , x2 , . . .) ∈ E∞ (X, T ) (with the same assumption x1 = x2 ). If k ∈ N : k
2, . µ µ µ then (x1 , . . . , xk ) ∈ Ek (X, T ) = supp(λk )\n and so x k = (x1 , . . . , xk , xk , . . .) ∈ supp(λ∞ ). µ µ k (∞) Since x → (x1 , x2 , . . .) as k → ∞ and supp(λ∞ ) is closed in X , (x1 , x2 , . . .) ∈ supp(λ∞ ). µ µ This implies that supp(λ∞ )\∞ = E∞ (X, T ). µ Moreover, select µ0 ∈ M(X, T ) such that En 0 (X, T ) = En (X, T ) for each n
2 (by µ proposition 2.4). To end the proof, it remains to show E∞ (X, T ) = E∞0 (X, T ), but this follows easily from theorem 2.7. This finishes the proof.  The next proposition is similar to proposition 2.5. Proposition 3.5. Let (X, T ) be a TDS and µ ∈ M(X, T ). Say µ = decomposition of µ. Then we have

 

µω dη(ω) is the ergodic

(1) For η-almost every ω ∈ , each entropy set for µω is an entropy set for µ. In particular, µ µ E∞ω (X, T ) ⊆ E∞ (X, T ) for η-almost every ω ∈ .   µ µω (2) If (x1 , x2 , . . .) ∈ E∞ (X, T ), then η({ω ∈ | ni=1 Vi × ∞ n+1 X ∩ E∞ (X, T ) = ∅}) > 0 for each n ∈ N : n
2 and any neighbourhood Vi of xi (1  i  n). Thus for an appropriate choice of  we can require µω µ cl(∪{E∞ (X, T ) : ω ∈ })\∞ = E∞ (X, T ). µ

µ

Proof. (1) By proposition 2.5, for η-almost every ω ∈ , En ω (X, T ) ⊆ En (X, T ) for all n ∈ N : n
2 and so each entropy set for µω is an entropy set for µ. In particular, one has µ µ E∞ω (X, T ) ⊆ E∞ (X, T ) for η-almost every ω ∈ . (2) Let n ∈ N : n
2 be fixed. Without loss of generality assume xi = xj for µ some 1  i < j  n. Then (x1 , . . . , xn ) ∈ En (X, T ), and so by proposition 2.5,

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 µ η({ω ∈ | ni=1 Vi ∩ En ω (X, T ) = ∅}) > 0. Note that for each ν ∈ M(X, T ) and ν ν (y1 , . . . , yn ) ∈ En (X, T ), (y1 , . . . , yn , yn , . . .) ∈ E∞ (X, T ). Thus 

 n ∞   µω η ω ∈ | Vi × X ∩ E∞ (X, T ) = ∅ i=1

n+1




η

ω ∈ |

n 

 Vi ∩

Enµω (X, T )

= ∅

> 0.

(3.5)

i=1 µ

In particular, (x1 , x2 , . . .) ∈ cl(∪{E∞ω (X, T ) : ω ∈ }). This means µ µω E∞ (X, T ) ⊆ cl(∪{E∞ (X, T ) : ω ∈ })\∞ .

Moreover, let  be any µ supp(λ∞ ) (using theorem 3.4)

µ appropriate choice satisfying E∞ω (X, T ) µ for each ω ∈ . Since supp(λ∞ ) ⊆ X(∞) is

(3.6) µ E∞ (X, T )

⊆ closed,

µω cl(∪{E∞ (X, T ) : ω ∈ })\∞ ⊆ supp(λµ∞ )\∞ .

⊆

(3.7) 

Thus the conclusion follows directly from theorem 3.4, (3.6) and (3.7).

Since each maximal entropy set has a countable dense subset, we have the following simple observation. Theorem 3.6. Let (X, T ) be a TDS and K
X be a subset of X with #(K)
2. Then the following statements are equivalent: (1) K is a maximal entropy set. (2) for any countable dense subset {xi }∞ 1 in K and any x ∈ X\K, (x1 , x2 , . . .) ∈ E∞ (X, T ), whereas (x, x1 , x2 , . . .) ∈ / E∞ (X, T ). (3) for any x ∈ X\K, there exists some countable dense subset {xi }∞ 1 in K such that (x1 , x2 , . . .) ∈ E∞ (X, T ), whereas (x, x1 , x2 , . . .) ∈ / E∞ (X, T ). 4. Some properties of maximal entropy sets In this section we shall show that if a TDS has positive entropy, then there is a maximal entropy set with uncountably many points and the entropy of the system is the supremum of the entropies over all maximal entropy sets. Let (X, T ) be a TDS, µ ∈ M(X, T ) and d be a metric on (X, T ). Put µ

A1 (µ) = {x1 ∈ X : there is x2 ∈ X\{x1 } with (x1 , x2 ) ∈ supp(λ2 )}, and inductively put for each n
2 An (µ) = {(x1 , . . . , xn ) ∈ X(n) : there is xn+1 ∈ X\{x1 , . . . , xn } µ with (xi )n+1 ∈ supp(λn+1 )}. 1 For each n
2, let pn : X(n) → X(n−1) be the map p n (x1 , . . . , xn ) = (x1 , . . . , xn−1 ). Then by the construction of An−1 (µ) one has An−1 (µ) = pn (supp(λµn ) ∩ {(x1 , . . . , xn ) ∈ X(n) : xn = xi if 1  i < n})  = p n (supp(λµn ) ∩ {(x1 , . . . , xn ) ∈ X(n) : min d(xn , xi )
1/m}); m∈N

1
i
n−1

this means that An−1 (µ) forms an Fσ subset of X(n−1) , and so it is ν-measurable for each ν ∈ M(X, T ). In particular, An (µ) (n ∈ N) are all µ-measurable. Roughly speaking, An (µ)\n is the set of entropy n-tuples for µ which can be extended to entropy (n + 1)tuples for µ.

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Lemma 4.1. Let (X, T ) be a TDS and µ ∈ M(X, T ) with hµ (T ) > 0. Suppose that µ =  µ dν(y) is the disintegration of µ over the Pinsker factor (Y, Dν , ν, S) of (X, Bµ , µ, T ), y Y where Bµ is the completion of BX under µ. Then (1) (2) (3) (4)

For ν-almost every y ∈ Y , supp(µy ) is an entropy set for µ if it is not a singleton. For ν-almost every y ∈ Y , supp(µy ) is an entropy set for µ when µ ∈ Me (X, T ). µ µ If µ ∈ Me (X, T ), then for each n
1, λn (An (µ)) = 1 and λn (An (µ)\n ) = 1. If K is an entropy set for µ, then K ⊂ supp(µ). µ

µ

Proof. First we show 1. Note that for all n
2, En (X, T ) = supp(λn )\n and  µ µ µ 1 = λn (supp(λn )) = µ(n) y (supp(λn ))dν(y) (by lemma 3.1). Y µ

(n) Thus for some Yn ⊆ Y , ν(Yn ) = 1 and µ(n) y (supp(λn )) = 1 if y ∈ Yn . Since supp(µy ) = µ µ (n) (n) (supp(µy )) and supp(λn ) is closed; one has (supp(µy )) ⊆ supp(λn ) for all y ∈ Yn . µ (n) Set Y∞ = n2 Yn . Then ν(Y∞ ) = 1, and (supp(µy )) \n ⊆ En (X, T ) when y ∈ Y∞ , n
2. Applying theorem 2.7, we have for all y ∈ Y∞ , supp(µy ) is an entropy set for µ if it is not a singleton. This ends the proof of 1. Now assume that µ ∈ Me (X, T ). Since hµ (T ) > 0, by the well-known Rohlin’s theorem for ν-almost every y ∈ Y , supp(µy ) is a compact metric space without isolated points. Then part 2 follows directly from part 1. Moreover, for each n
1, supp(µy )(n) ⊆ An (µ) ν-almost every y ∈ Y . Thus   (n) 1
λµn (An (µ))
µ(n) (A (µ))dν(y)
µ(n) n y y (supp(µy ) )dν(y) = 1. Y µ λn (An (µ))

Y

µ λn (An (µ)\n )

µ

We have = 1, and thus = 1 since λn (n ) = 0. µ To end the proof we assume that K is an entropy set for µ. Then K ×K\2 ⊂ E2 (X, T ) ⊂ µ µ supp(λ2 ). This implies K ⊂ supp(µ) by the definition of λ2 .  Applying the variational principle to lemma 4.1, we have Theorem 4.2. Let (X, T ) be a TDS with positive entropy. Then there exists a maximal entropy set with uncountably many points. As a corollary of lemma 4.1 we have Corollary 4.3. Under the same assumption of lemma 4.1, we have the following: there exists Y∞ ⊆ Y with ν(Y∞ ) = 1 and Y0 = {y1 , y2 , . . .} ⊆ Y∞ such that for each n
2 supp(λµn )

=

 y∈Y∞

(supp(µy

))(n)

=

∞ 

(supp(µyi ))(n) .

i=1

Proof. Let Y∞ be the measurable subset constructed in lemma 4.1. Then ν(Y∞ ) = 1, and . µ (supp(µy ))(n) ⊆ supp(λn ) when y ∈ Y∞ , n
2. Thus Wn = y∈Y∞ (supp(µy ))(n) ⊆ µ µ supp(λn ), for supp(λn ) is closed in X(n) . On the other hand,   (n) λµn (Wn ) = µ(n) (W )dν(y)
µ(n) n y y ((supp(µy )) )dν(y) = 1. Y µ supp(λn ).

Y∞

Thus Wn = Moreover, for any fixed n
2, choose {y1n , y2n , . . .} ⊆ Y∞ such that

∞

Wn = i=1 (supp(µyin ))(n) . Set Y0 = n2 {y1n , y2n , . . .}, rewrite it as Y0 = {y1 , y2 , . . .} ⊆ Y∞ .

(n) for each n
2. This finishes our proof.  One has Wn = ∞ i=1 (supp(µyi ))

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In the topological setting, let (X, T ) be a TDS and put A1 = {x1 ∈ X : there is x2 ∈ X\{x1 } with (x1 , x2 ) ∈ E2 (X, T )}, and inductively put for each n
2 An = {(x1 , . . . , xn ) ∈ X(n) : there is xn+1 ∈ X\{x1 , . . . , xn } with (xi )n+1 ∈ En+1 (X, T )}. 1 Then for each n
2 one has An−1 = pn (En (X, T ) ∩ {(x1 , . . . , xn ) ∈ X(n) : xn = xi if 1  i < n})  = p n (En (X, T ) ∩ {(x1 , . . . , xn ) ∈ X(n) : min d(xn , xi )
1/m}), 1
i
n−1

m∈N

which implies An−1 is an Fσ subset of X (n−1) , and so it is ν-measurable for each ν ∈ M(X, T ). Roughly speaking, An \n is the set of topological entropy n-tuples which can be extended to topological entropy (n + 1)-tuples. Let µ ∈ M(X, T ). It is clear An (µ) ⊆ An for n ∈ N. Thus lemma 4.1 tells us that µ if µ ∈ Me (X, T ) satisfies hµ (T ) > 0, then λn (An ) = 1 for each n ∈ N. Moreover, by µ proposition 2.4 we can choose µ0 ∈ M(X, T ) such that En 0 (X, T ) = En (X, T ) for all n
2, and thus An (µ0 ) = An when n
2. In the rest of the section we shall discuss entropies over maximal entropy sets. Before proceeding, let us first recall the definition of topological entropy using Bowen’s definition of separated and spanning sets and some relevant results (see [W], P168–174 ). Let (X, T ) be a TDS with d a metric on (X, T ). For n ∈ N we define a new metric dn on the space by dn (x, y) = max0
i
n−1 d(T i x, T i y). Let  > 0 and K ⊂ X. A subset F of X is said to (n, ) span K with respect to T if ∀x ∈ K, ∃y ∈ F with dn (x, y)  , a subset E of K is said to be (n, ) separated with respect to T if x, y ∈ E, x = y implies dn (x, y) > . Let rn (d, T , , K) denote the smallest cardinality of any (n, )-spanning set for K with respect to T and sn (d, T , , K) denote the largest cardinality of any (n, ) separated subset of K with respect to T . Put 1 log rn (d, T , , K), n→∞ n 1 s(d, T , , K) = lim sup log sn (d, T , , K). n→∞ n

r(d, T , , K) = lim sup

Obviously, rn (d, T , , K) < ∞ and rn (d, T , 1 , K)
rn (d, T , 2 , K) when 1 < 2 . The same argument works for sn (d, T , , K). Then set h∗ (d, T , K) = lim r(d, T , , K) and h∗ (d, T , K) = lim s(d, T , , K). →0+

→0+

Moreover, it is not hard to obtain the following easy facts: (1) h∗ (d, T , K) = h∗ (d, T , K), thus we shall denote them by h(d, T , K). Moreover, h(d, T , K) depends only on the topology on X. (2) htop (T ) = h(d, T , X). Lemma 4.4. Let (X, T ) be a TDS with d the metric on (X, T ). If µ ∈ Me (X, T ) satisfies hµ (T ) > 0, then hµ (T )  sup{h(d, T , K) : K is an entropy set for µ}. Proof. We shall follow the idea of theorem 4.4 in [Z] (see also theorem 8.6  in [W]). Let (Y, D, ν, S) be the Pinsker factor of (X, Bµ , µ, T ) and µ = Y µy dν(y) be the disintegration of µ over ν, where Bµ is the completion of BX under µ. Then lemma 4.1 . tells us that for ν-almost every y ∈ Y , Sy = supp(µy ) is an entropy set for µ.

Entropy sequences and maximal entropy sets

65

Let essupy∈Y denote the supremum taken over ν-almost every y ∈ Y , that is to say essupy∈Y =

inf

sup .

E⊆Y :ν(E)=0 y∈Y \E

Then we only need to prove the inequality hµ (T )  essupy∈Y h(d, T , Sy ). Let ξ = {A1 , . . . , As } be any finite Borel partition of X. Choose compact sets B

i ⊆ Ai (1  i  s) such that Hµ (ξ |γ )  1, where γ = {B0 , B1 , . . . , Bs } and B0 = X\ si=1 Bi (see [W], P189 ). Then we have hµ (T , ξ )  Hµ (ξ |γ ) + hµ (T , γ ) (by the Pinsker formula)  1 + hµ (T , γ )  1 Hµy (γ0n−1 )dν(y) (by (3.1)) = 1 + lim n→∞ n Y  1  1 + lim sup Hµy (γ0n−1 )dν(y) (by the Bounded Convergence theorem) Y n→∞ n 1  1 + essupy∈Y lim sup Hµy (γ0n−1 ). (4.1) n→∞ n For n ∈ N, ν-almost every y ∈ Y , let Ay be the set of all atoms of γ0n−1 which has nonempty intersection with Sy and β(y) the cardinality of Ay . Clearly, Hµy (γ0n−1 )  log β(y). Let U be the open cover {B0 ∪ B1 , . . . , B0 ∪ Bs } and δ2 a Lebesgue number of U . Choose a positive number δ3 < 41 δ2 . Claim. β(y)  2n rn (d, T , δ3 , Sy ). Proof of claim. Let Ey be any (n, δ3 )-spanning set for Sy with respect to T with the minimal cardinality rn (d, T , δ3 , Sy ). For any element C of Ay , fix x(C) ∈ C ∩ Sy , since Ey is an (n, δ3 )-spanning Sy with respect to T , we can choose y(C) ∈ Ey such that dn (x(C), y(C))  δ3 . Note that δ2 is a Lebesgue number of U and δ2 > 4δ3 , then clB(T i x(C), 2δ3 ) is contained in some element of U , say clB(T i x(C), 2δ3 ) ⊆ B0 ∪ Bji , i = 0, 1, . . . , n − 1. If C1 , . . . , Cq (q ∈ N) are other q-elements of Ay satisfying y(C) = y(C1 ) = · · · = y(Cq ), then dn (x(Ct ), y(Ct ))  δ3 and so dn (x(C), x(Ct ))  2δ3 (1  t  q). In particular, T i x(C), T i x(C1 ), . . . , T i x(Cq ) ∈ B0 ∪ Bji , i = 0, 1, . . . , n − 1. That is, x(C1 ), . . . , x(Cq ) and x(C) belong to the same element
−i of n−1 i=0 T U , say W0 , so Ct ∩ W0 = ∅ (1  t  q) and C ∩ W0 = ∅.
−i Note that for any atom F of γ0n−1 and each element W of n−1 i=0 T U , say F = n−1 −i n−1 −i i=0 T Bki and W = i=0 T (B0 ∪ Bli ) (0  ki  s and 1  li  s when 0  i  n − 1), by the definition of γ , if F ∩ W = ∅, then Bki ∩ (B0 ∪ Bli ) = ∅ and so Bki ⊆ (B0 ∪ Bli ). That is, F ∩ W = ∅ implies F ⊆ W . This means C1 , . . . , Cq and C are all contained in W0 .
−i n According to the definitions of γ and U , each element of n−1 i=0 T U contains at most 2 n−1 n atoms of γ0 , then there are at most 2 elements of Ay with y(C) equal to a fixed point of Ey . Hence the cardinality of Ay is at most 2n times of the cardinality of Ey . That is, β(y)  2n rn (d, T , δ3 , Sy ). This ends the proof of the claim.  By the claim we have Hµy (γ0n−1 )  log β(y)  n log 2 + log rn (d, T , δ3 , Sy ). Thus by (4.1) and (4.2) one has hµ (T , ξ )  1 + log 2 + essupy∈Y r(d, T , δ3 , Sy )  1 + log 2 + essupy∈Y h(d, T , Sy ). Moreover, hµ (T )  1 + log 2 + essupy∈Y h(d, T , Sy ).

(4.2)

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For m ∈ N and  > 0, since rmn (d, T , , Sy )
rm (d, T n , , Sy ), we have 1 1 log rmn (d, T , , Sy )
h(d, T n , Sy ). (4.3) h(d, T , Sy )
lim lim sup →0+ m→∞ mn n Now apply the above discussion to T n instead of T . By (4.3) one has nhµ (T )  1 + log 2 + n · essupy∈Y h(d, T , Sy ), dividing by n and letting n → ∞, to deduce hµ (T )  essupy∈Y h(d, T , Sy ). This completes the proof of the lemma.  Thus applying the variational principle to lemma 4.4, we have the following. Theorem 4.5. Let (X, T ) be a TDS with d the metric on (X, T ). Then htop (T ) = sup{h(d, T , K) : K is an entropy set} = sup{h(d, T , K) : K is an entropy set for some invariant measure} = sup{h(d, T , K) : K is an entropy set for some ergodic invariant measure}. We remark that it is easy to construct a TDS (X, T ) such that there is no maximal entropy set K with h(T , K) = htop (T ). For example, let (Xi , Ti ) be a sequence of disjoint topological K systems embedded in R with log 2 > htop (Ti ) → log 2, diam(Xi ) → 0 and Xi → {0} in the Hausdorff metric. If we let X be the union of all Xi with {0} and define a map T with T |Xi = Ti and T (0) = 0, then (X, T ) has no maximal entropy set with entropy log 2. Moreover, in a paper being prepared by Ye and Zhang [YZ] it is shown that for any TDS (X, T ) with d a metric on X there is a compact countable subset K with h(d, T , K) = htop (T ). Thus, theorem 4.2 is not a direct consequence of theorem 4.5. A corollary of lemma 3.1 or theorem 4.5 is the following. Corollary 4.6. Let (X, T ) be a TDS. Then, for each n
2, htop (T (n) |En (X,T ) ) = nhtop (T ). We remark that if (X, T ) is a TDS and µ ∈ Me (X, T ) with hµ (T ) > 0, generally µ µ × µ is not a probability measure on E2 (X, T ). It is not known (or it is an open question) µ whether µ × µ(E2 (X, T )) > 0. The following result is notable in ergodic theory (for example theorem 8.4 of [W]).  Lemma 4.7. Let (X, T ) be a TDS and µ ∈ M(X, T ). Say µ =  µω dη(ω) is the ergodic decomposition of µ. Then we have  (1) For any given  finite Borel partition α of X, holds hµ (T , α) =  hµω (T , α)dη(ω). (2) hµ (T ) =  hµω (T )dη(ω). With the help of lemma 4.7 we can show the following. Theorem 4.8. Let (X, T ) be a TDS with d the metric on (X, T ). If µ ∈ M(X, T ) satisfies hµ (T ) > 0. Then one has hµ (T )  sup{h(d, T , K) : K is an entropy set for µ}.  Proof. Let µ =  µω dη(ω) be the ergodic decomposition of µ. By proposition 2.5, then for µ µ some 0 with η(0 ) = 1 one has En ω (X, T ) ⊆ En (X, T ) for each n
2 and ω ∈ 0 . Thus if ω ∈ 0 , any entropy set for µω is also an entropy set for µ, and so  hµω (T )dη(ω) (by lemma 4.7)  sup hµω (T ) hµ (T ) = 0

ω∈0

 sup sup{h(d, T , Kω ) : Kω is an entropy set for µω } (by lemma 4.4) ω∈0

 sup sup{h(d, T , K) : K is an entropy set for µ} ω∈0

= sup{h(d, T , K) : K is an entropy set for µ}. This proves the inequality.



Entropy sequences and maximal entropy sets

67

5. Systems with only one maximal entropy set In this section we shall characterize systems with only one maximal entropy set. Note that it is easy to construct a TDS with several maximal entropy sets, for example, the one in the remark after theorem 4.5. To have a transitive example, we take (X, T ) to be the full shift on two symbols and (Y, S) to be a transitive zero entropy non-trivial TDS and form the product system (Z, W ) which is transitive. It is easy to show that each maximal entropy set of (Z, W ) has a form of X × {y} with y ∈ ∪ν∈M(Y,S) supp(ν). To start our study first we will discuss the case when there is a unique maximal entropy set with respect to an invariant measure µ. Let (X, T ) be a TDS and µ ∈ M(X, T ) with hµ (T ) > 0. Set Xµe to be the smallest closed subset which contains all entropy sets for µ. Then Xµe is T -invariant, and (Xµe , T ) is a non-trivial TDS. Say µ =  µω dη(ω) is the ergodic decomposition of µ. Then 0 = {ω ∈  : hµω (T ) > 0} ⊂  is measurable. Moreover, since hµ (T ) > 0, η(0 ) > 0. In this case, . set µe = (1/η(0 )) 0 µω dη(ω). For example, µe = µ when µ ∈ Me (X, T ). Then µ = η(0 )µe + η(\0 )µr ,  where µr = (1/η(\0 )) \0 µω dη(ω) when η(0 ) < 1 and µr = µ when η(0 ) = 1. Note that µe , µr ∈ M(X, T ) are both determined completely by µ. Proposition 5.1. Let (X, T ) be a TDS and µ ∈ M(X, T ) with hµ (T ) > 0. Then µe

µ

(1) En (X, T ) = En (X, T ) for each n
2. (2) Xµe = supp(µe ) and hµe (T |Xµe )
hµ (T ). (3) We have  ∪{supp(µω ) : ω ∈ ∗ } Xµe = ∗ ⊂

0

:η(∗ )=η(

and

hµ (T ) = η(0 )hµe (T |Xµe ).

0)

 . Proof. Let µ =  µω dη(ω) be the ergodic decomposition of µ and put 0 = {ω ∈  : hµω (T ) > 0}. For any given finite Borel partition α of X, by lemma 4.7 we have   hµ (T , α) = hµω (T , α)dη(ω) = hµω (T , α)dη(ω) = η(0 )hµe (T , α). 

0

Since η(0 ) > 0, part 1 follows easily from simple discussions. Moreover, hµ (T ) = η(0 )hµe (T ). Since Xµe = supp(µe ) (to be proved later), µe can be viewed as an invariant probability measure on TDS (Xµe , T ); one has hµ (T ) = η(0 )hµe (T ) = η(0 )hµe (T |Xµe ). To prove the proposition, it remains to show supp(µe ) = Xµe = X(µ), where  . X(µ) = ∪{supp(µω ) : ω ∈ ∗ }. ∗ ⊂0 :η(∗ )=η(0 )

We divide the proof into the following steps. Step 1. supp(µe ) ⊇ Xµe . It follows easily from part 1, theorem 2.7 and lemma 4.1 (4). Step 2. Xµe ⊇ X(µ). By proposition 2.5 and theorem 2.7, there is some measurable  ⊆  with η( ) = 1 such that ∀ω ∈  , each entropy set for µω is also an entropy set for µ. Setting

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1 = 0 ∩  , we get η(1 ) = η(0 ). To complete the step, we claim that Xµe ⊇ X1 , where . X1 = ∪{supp(µω ) : ω ∈ 1 } ⊇ X(µ). Let ω ∈ 1 . Say µω = Yω µy,ω dνω (y) is the disintegration of µω over the Pinsker factor of µω . Then by lemma 4.1, for νω -almost every y ∈ Yω , supp(µy,ω ) is an entropy set for µω and so for µ, thus supp(µy,ω ) ⊆ Xµe . Since µω (Xµe ) = Yω µy,ω (Xµe )dνω (y) = 1, we conclude that Xµe ⊇ supp(µω ). Since Xµe is a closed subset, Xµe ⊇ X1 , which ends the proof of claim. Step 3. X(µ) ⊇ supp(µe ). Let ∗ ⊂ 0 be a measurable subset with η(∗ ) = η(0 ). Set X∗ = ∪{supp(µω ) : ω ∈ ∗ }. By the construction of µe , we have   1 1 e ∗ ∗ µ (X ) = µω (X )dη(ω) = µω (X ∗ )dη(ω) = 1. µ(0 ) 0 µ(0 ) ∗ Thus X∗ ⊇ supp(µe ). Moreover, X(µ) ⊇ supp(µe ).



Now we give a characterization of a TDS with a unique maximal entropy set for an invariant measure. Theorem 5.2. Let (X, T ) be a TDS and µ ∈ M(X, T ) with hµ (T ) > 0. Suppose that K is the unique maximal entropy set for µ in (X, T ). Then K = Xµe and (K, T ) forms a non-trivial topological K sub-system of (X, T ). Moreover, htop (T |K )
hµ (T ). Proof. By proposition 5.1, obviously K = Xµe = supp(µe ) is non-trivial. In particular, µe is supported in (K, T ), and thus µe can be viewed as an invariant probability measure on (K, T ). Since K is an entropy set for µ, for each n
2 one has by proposition 5.1 e

e

K (n) \n (K) ⊆ Enµ (X, T ) = Enµ (X, T ) = Enµ (K, T ) ⊆ En (K, T ). Thus En (K, T ) = K (n) \n (K) when n
2, and so (K, T ) is topological K. Moreover, htop (T |K )
hµ (T ) follows from theorem 4.8. This ends the proof.  Now let us turn to the topological setting. Let (X, T ) be a TDS with positive entropy. Set Xe to be the smallest closed subset which contains all entropy sets. Then for any given µ ∈ M(X, T ), Xµe ⊆ X e , and we can select some µ0 ∈ M(X, T ) such that µ En 0 (X, T ) = En (X, T ) for all n
2 (proposition 2.4) which implies Xµe 0 = Xe by theorem 2.7. Obviously, Xe is T -invariant and (X e , T ) is non-trivial. Moreover, X e can be characterized in the following way. Proposition 5.3. Let (X, T ) be a TDS with positive entropy. Then Xe = ∪{supp(µ) : hµ (T ) > 0 and µ ∈ Me (X, T )}

and

htop (T |Xe ) = htop (T ).

Proof. The part of htop (T |Xe ) = htop (T ) follows easily from theorem 4.5. Now set X0 = ∪{supp(µ) : hµ (T ) > 0 and µ ∈ Me (X, T )}. It remains to prove X e = X0 . On the one hand, for any µ ∈ Me (X, T ) with hµ (T ) > 0, since µe = µ we have e X ⊇ Xµe = supp(µ) by proposition 5.1 and so X e ⊇ X0 . On the other hand, let µ ∈ M(X, T ) satisfy X e = Xµe . Then µe =  (1/η(0 )) 0 µω dη(ω), where µω is ergodic and hµω (T ) > 0. Thus, supp(µω ) ⊆ X0 if  ω ∈ 0 . Since µe (X0 ) = (1/η(0 )) 0 µω (X0 )dη(ω) = 1, we have X0 ⊃ supp(µe ). It follows by proposition 5.1, Xe = Xµe = supp(µe ) ⊆ X0 , and thus X e = X0 .  Now we give a characterization of a TDS with a unique maximal entropy set.

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Theorem 5.4. Let (X, T ) be a TDS with positive entropy. Then (X, T ) has a unique maximal entropy set if and only if X e is an entropy set. Moreover, when (X, T ) admits a unique maximal entropy set K, we have K = X e , (K, T ) is the unique maximal non-trivial topological K subsystem of (X, T ) and htop (T |K ) = htop (T ). Proof. First assume that (X, T ) admits a unique maximal entropy set K. By the definition K ⊂ X e . At the same time, if K  is an entropy set then K  ⊂ K, and thus Xe ⊂ K. That is, Xe = K is an entropy set. Conversely assume that X e is an entropy set. As each entropy set is contained in Xe , X e is the unique maximal entropy set. When (X, T ) admits a unique maximal entropy set K, it is clear that K = Xe is non-trivial and htop (T |K ) = htop (T ) (theorem 4.5 or proposition 5.3). µ By proposition 2.4 we can select µ ∈ M(X, T ) such that En (X, T ) = En (X, T ) for all n
2. Then K is the unique maximal entropy set for µ in (X, T ). Thus theorem 5.2 tells us that the sub-system (K, T ) is topological K. Let (X1 , T ) be another topological K non-trivial sub-system of (X, T ); then X1 is an entropy set for (X1 , T ) and so for (X, T ). Thus by assumption we have X1 ⊆ K. That is, (K, T ) forms the unique maximal topological K sub-system of (X, T ).  The obvious TDS having a unique maximal entropy set is the topological K system. A system that is not topological K but has a unique maximal entropy set can be obtained by taking the disjoint union of a topological K system and a system with zero entropy. To get a transitive example is a much harder task (though we believe that one exists). 6. Possible cardinality of a maximal entropy set In section 4 we showed that if a TDS has positive entropy then it has a maximal entropy set with uncountably many points. In this last section we shall study the other possibility of the cardinality of a maximal entropy set. Namely, we will construct an example with a maximal entropy set consisting of only two points. We believe that by modifying the construction of our example for any given n
3 there is a TDS with a maximal entropy set consisting of n points, and there is a TDS with a maximal entropy set consisting of countable infinitely many µ µ points. Note that when µ ∈ Me (X, T ) with hµ (T ) > 0, λn is ergodic and λn (An ) = 1 for each n ∈ N. To do this, we need some preparations. Definition 6.1. Let (X, T ) be a TDS and U0 , U1 be two non-empty open subsets of X. We say that (X, T ) has property P with respect to U0 and U1 if there is N > 0 such that for any k
2 and each s = (s(1), . . . , s(k)) ∈ {0, 1}k , there exists y ∈ X with y ∈ Us(1) , . . . , T (k−1)N (y) ∈ Us(k) . It is easy to prove the following lemma (see for example [HY]). Lemma 6.2. Let (X, T ) be a TDS and U0 , U1 be two open subsets of X satisfying U0 ∩U1 = ∅. Suppose that (X, T ) has property P with respect to U0 and U1 , and R = {U, V } is an open cover of X where U0 ⊂ U c and U1 ⊂ V c . Then htop (T , R) > 0. For p
2, let  = {0, 1, . . . , p − 1} be a finite set of symbols equipped with the discrete topology, and  = N be the set of all one-sided infinite sequences (xn )n∈N of elements of  equipped with the product topology. Let σ :  →  be the shift defined by σ ((xn )n∈N ) = (xn+1 )n∈N . We say a TDS (X, σ ) is a subshift on  if X is a closed σ -invariant subset of . For n
2 and a = (a1 , a2 , . . . , an ) ∈ n (a block of length n), set |a| = n,

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σ (a) = (a2 , . . . , an ). We say that a appears in x = (x1 , x2 , . . .) or x = (x1 , . . . , xm ) ∈ m , where m
n if there is j ∈ N such that a = (xj , xj +1 , . . . , xj +n−1 ). In this case, we write a < x for short. When b = (b1 , . . . , bm ) ∈ m , we define ab = (a1 , . . . , an , b1 , . . . , bm ) ∈ n+m , and we shall use t i to denote t · · · t (i-times). Let (X, σ ) be a subshift on  and a = (a1 , . . . , an ) ∈ n (n ∈ N). Put [a1 , . . . , an ] = {y = (y1 , y2 , . . .) ∈ X : (y1 , . . . , yn ) = (a1 , . . . , an )}. If 1  m  n, then for each 1  i  n + 1 − m put (a)[i,i+m−1] = [ai , . . . , ai+m−1 ]. Assume U = {U0 , U1 , . . . , Uq−1 } is an open cover of X, and K ⊆ {0, 1, . . . , q − 1}n (n ∈ N). We say that K (U , l)-covers X if  X= Ui0 ∩ σ −l Ui1 ∩ · · · ∩ σ −(n−1)l Uin−1 . (i0 ,...,in−1 )∈K

In this case, each k ∈ K is called a (U , l)-name of length n. In the following, we shall construct a transitive subshift (X, σ ) on {0, 1, 2} such that x = (x1 , x2 , . . .) is a transitive point of (X, σ ) satisfying (x, σ (x)) ∈ E2 (X, σ ); y 1 = (0, 1, 0, 1, . . .) ∈ X and (y 1 , y 2 , y 3 ) ∈ / E3 (X, σ ) for any point y 3 ∈ X\{y 1 , y 2 }, where 2 1 y = σy . The idea of the construction (inspired by theorem 9.4 in [HY]) can be explained as follows: we construct a recurrent point x = (x1 , x2 , . . .) such that (1) for each closed neighbourhood U, V of x and σ x the open cover {U c , V c } has positive entropy (this is done by putting in x many different blocks and using lemma 6.2); (2) for each n, the block (0101 . . . 01) of length 2n appears in x which implies that y 1 , y 2 ∈ E2 (X, σ ); (3) for each y 3 ∈ X\{y 1 , y 2 } and some disjoint closed neighbourhoods Ui of y i , the open cover {U1c , U2c , U3c } has zero entropy (this is done by estimating that the growth rate of the open cover is polynomial and implies that (y 1 , y 2 , y 3 ) ∈ / E3 (X, σ )). As the example is not invertible we modify it to get an invertible one. Theorem 6.3. There exists a transitive TDS (Y, S) with positive entropy such that one of its maximal entropy sets contains only two points. Proof. We shall first construct a non-invertible transitive one-sided subshift (X, σ ) ⊆ {0, 1, 2}N such that x = (x1 , x2 , . . .) is a transitive point of (X, σ ) and (1) (x, σ (x)) ∈ E2 (X, σ ) and y 1 = (0, 1, 0, 1, . . .) ∈ X. (2) (y 1 , y 2 , y 3 ) ∈ / E3 (X, σ ) for any point y 3 ∈ X\{y 1 , y 2 } where y 2 = σy 1 . Let φ : N → N be a map such that φ −1 (j ) is infinite for each j ∈ N. Let A1 = (2222). Set n1 = |A1 |, C01 = A1 (01)n1 /2 and C11 = σ (A1 )0(01)n1 /2 . Assume 1 {D11 · · · Dn11 , Dn11 +1 · · · D2n , . . . , Dn11 2n1 −n1 +1 · · · Dn11 2n1 } = {C01 , C11 }n1 , 1

where Dj1 ∈ {C01 , C11 } (1  j  n1 2n1 ). Put 1 A2 = A1 (01)n1 /2 D11 · · · Dn11 Dn11 +1 · · · D2n · · · Dn11 2n1 −n1 +1 · · · Dn11 2n1 , 1

and n2 = |A2 |. If A1 , . . . , Ak are defined, set C0k = Aφ(k) (01)nφ(k) /2 and C1k = σ (Aφ(k) )0(01)nφ(k) /2 . Assume k {D1k · · · Dnkk , Dnkk +1 · · · D2n , . . . , Dnkk 2nk −nk +1 · · · Dnkk 2nk } = {C0k , C1k }nk , k

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where Djk ∈ {C0k , C1k } (1  j  nk 2nk ). Put k · · · Dnkk 2nk −nk +1 · · · Dnkk 2nk , nk+1 = |Ak+1 |. Ak+1 = Ak (01)nk /2 D1k · · · Dnkk Dnkk +1 · · · D2n k

Then nk+1 ≡ 0 (mod 2nk ) and nk+1 > 4nk for k ∈ N. Let X = {x, σ (x), σ 2 (x), . . .} ⊆ {0, 1, 2}N where x = limk→∞ Ak . Since there exists a sequence {sj }j ∈N ⊆ N such that sj  +∞ and σ sj (x) → x, (X, σ ) is transitive. We claim that (X, σ ) is the system we need. Step 1. First we show (x, σ (x)) ∈ E2 (X, σ ). Let U be a neighbourhood of x and V be a neighbourhood of σ (x). By the definition of x, there is k ∈ N such that U0 = [Ak ] ⊆ U and V0 = [σ (Ak )0] ⊆ V . As Ak = σ (Ak )0, U0 ∩ V0 = ∅, and so V = {U0c , V0c } forms an open cover of (X, σ ). Since there are infinitely many j such that φ(j ) = k, we have j

j

C0 = Ak (01)nk /2 , nj /2

j D1

j

j nj

Aj +1 = Aj (01) j

j

C1 = σ (Ak )0(01)nk /2 j · · · Dnj j Dnj +1

where Dlnj +1 · · · D(l+1)nj ∈ {C0 , C1 } j

j · · · D2nj

and

j · · · Dnj 2nj −nj +1

j

· · · Dnj 2nj ,

for l = 0, 1, . . . , 2nj − 1.

j

Thus [C0 ] ⊆ U0 and [C1 ] ⊆ V0 for infinitely many j . Moreover, (X, σ ) has property P with respect to U0 and V0 (with N = 2nk ). This implies htop (σ, V ) > 0 (by lemma 6.2). Consequently, if U ∩ V = ∅ then htop (σ, {U c , V c }) > 0, as in this case {U c , V c } is an open cover of X which is finer than V . That is, (x, σ (x)) ∈ E2 (X, σ ). It is obvious that y 1 , y 2 ∈ X. Since x is a transitive point of (X, σ ), we have (y, σ (y)) ∈ E2 (X, T ) if y ∈ X is not a fixed point. This implies that (y 1 , y 2 ) ∈ E2 (X, σ ). / E3 (X, σ ). Step 2. Let y 3 ∈ X\{y 1 , y 2 } be fixed. We now prove (y 1 , y 2 , y 3 ) ∈ Say y i = (y1i , y2i , . . .) (i = 1, 2, 3). Then there exists k ∈ N (fixed) such that / [y13 , . . . , yn3k ]. Set Ui = [y1i , . . . , yni k ] (i = 1, 2, 3) and U = {U1c , U2c , U3c }. We y , y2 ∈ / E3 (X, σ ). shall prove that htop (U , σ ) = 0 which implies that (y 1 , y 2 , y 3 ) ∈ For convenience write 1

j

j

j

j

j

Dj = D1 · · · Dnj j Dnj +1 · · · D2nj · · · Dnj 2nj −nj +1 · · · Dnj 2nj , ∀j ∈ N; α(i0 , . . . , in−1 ) = Uic0 ∩ σ −nk Uic1 ∩ · · · ∩ σ −(n−1)nk Uicn−1 , ∀{i0 , . . . , in−1 } ∈ {1, 2, 3}n . Note that when ri , si ∈ {0, 1, 2}, 0  i  nk − 1, there is a j ∈ {1, 2, 3} such that [r0 , r1 , . . . , rnk −1 ] ∪ [s0 , s1 , . . . , snk −1 ] ⊆ Ujc . Then when a, b ∈ {{0, 1, 2}nk }n , say a = (a1 , . . . , an ) and b = (b1 , . . . , bn ), there is c = (c1 , . . . , cn ) ∈ {1, 2, 3}n such that [a1 , . . . , an ] ∪ [b1 , . . . , bn ] ⊆ α(c1 , . . . , cn ).

(6.1)

Let r > k, and A be a block of length nr which appears in x. By the definition of x, there exists d
r such that A appears in Ad . Then doing induction on d
r, we have that one of the following cases holds by considering the structure of the block Ad : (1) (2) (3) (4)

A < Ar (01)nr /2 , (01)nr /2 Ar , σ (Ar )0(01)nr /2 , (01)nr /2 σ (Ar )0 or (10)nr /2 (01)nr /2 . A < Dr  , where r   d satisfies φ(r  ) < r  r  . A ≮ Dr  , but A < (01)nr /2 Dr  or Dr  (01)nr /2 , where r   d satisfies φ(r  ) < r  r  . A ≮ Dr  (01)nr /2 , but A < σ (Dr  )0(01)nr /2 , where r   d satisfies φ(r  ) < r  r  .

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Denote by Yi the union of cylinders [A] over all such blocks A of length nr which appear in the ith case (i = 1, 2, 3, 4). Case 1. The block A has no more than 5(nr + 1) choices appearing in this case, and so Y1 can be (U , nk )-covered by no more than 5(nr + 1) many (U , nk )-names. 





Case 2. Let l = φ(r  ) < r. Since Djr ∈ {C0r , C1r } = {Al (01)nl /2 , σ (Al )0(01)nl /2 } (1  j  nr  2nr  ), we have the following two subcases. 



(1) l < k. Since |C0r | = |C1r | = 2nl  nk /2, one has (10)nk /2 ≮ Dr  . In this case, Y2 can be (U , nk )-covered by a single (U , nk )-name 2nr /nk .   (2) k  l < r. Since |C0r | = |C1r | = 2nl
2nk , one has by (6.1) that there exists i,l i,l i,l (c1 , c2 , . . . , c(2nl /nk )−1 ) ∈ {1, 2, 3}(2nl /nk )−1 (1  i  nk ) such that 



i,l ). (C0r )[i,2nl −1−nk +i] ∪ (C1r )[i,2nl −1−nk +i] ⊆ α(c1i,l , c2i,l , . . . , c(2n l /nk )−1 

(6.2)



Note that for each E ∈ {C0r , C1r }2 , (E)[2nl −nk +i,2nl +i−1] ⊆ α(2) (1  i  nk ); thus in this case, Y2 can be (U , nk )-covered by the following set of (U , nk )-names (using (6.2))   2nl i,l nr /2nl , (6.3) , 2)) : 1  i  n , 0  t < (P t (c1i,l , c2i,l , . . . , c(2n k l /nk )−1 nk where P (a) = (a2 , . . . , an , a1 ) if a = (a1 , a2 , . . . , an ) ∈ n (n
2). To sum up, Y2 can be (U , nk )-covered by no more than 1 + nk (r − k)

2nl = 1 + 2nl (r − k)  1 + 2nr (r − k) nk

many (U , nk )-names. Case 3. Let l = φ(r  ) < r. By the same arguments as in case 2, we have (1) If l < k, then Y3 can be (U , nk )-covered by the following set of (U , nk )-names   nr n (nr /nk )−n (nr /nk )−n n . ), (2 ,3 ) : 0  n  (3 , 2 nk

(6.4)

(2) If k  l < r, then Y3 can be (U , nk )-covered by the following set of (U , nk )-names  i,l i,l , 2)m , c1i,l , . . . , c(n , (3n , 2, (c1i,l , . . . , c(2n l /nk )−1 r −2nl m−nk (n+1))/nk ) i,l i,l i,l m n (c(2n , . . . , c(2n , (2, c1i,l , . . . , c(2n /n )−1 ) , 3, 3 ) : l /nk )−((nr −2nl m−nk (n+1))/nk ) l /nk )−1 l k  nr nr − nk (n + 1) 1  i  nk , 0  n  , m = , nk 2nl

(6.5)

where [(nr − nk (n + 1))/2nl ] denotes the integral part of (nr − nk (n + 1))/2nl . Thus from (6.4) and (6.5), Y3 can be (U , nk )-covered by no more than     nr nr + 1 + 2nk (r − k) + 1  2(nr + nk )(r − k + 1) 2 nk nk many (U , nk )-names. Case 4. Let l = φ(r  ) < r. By similar arguments we have (1) If l < k, then Y4 can be (U , nk )-covered by the set {(2(nr /nk )−n , 3n ) : 0  n  (nr /nk )} of (U , nk )-names.

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(2) If k  l < r, by (6.1) there exists (e1i,l , e2i,l ) ∈ {1, 2, 3}2 (1  i  nk ) such that 



(C0r 0(01)nk /2 )[2nl −nk +i,2nl +nk +i−1] ∪ (C1r 0(01)nk /2 )[2nl −nk +i,2nl +nk +i−1] ⊆ α(e1i,l , e2i,l ). (6.6) Then using (6.6), Y4 can be (U , nk )-covered by the following set of (U , nk )-names  i,l i,l i,l , . . . , c(2n , (2, c1i,l , . . . , c(2n )m1 , e1i,l ), (c(2n l /nk )−((nr −2nl m1 −nk )/nk ) l /nk )−1 l /nk )−1 i,l i,l i,l (c(2n , . . . , c(2n , (2, c1i,l , . . . , c(2n )m2 , e1i,l , e2i,l , 3n ) : l /nk )−((nr −2nl m2 −nk (n+2))/nk ) l /nk )−1 l /nk )−1     nr nr − nk nr − nk (n + 2) , m2 = , (6.7) 1  i  nk , 0  n  , m1 = nk 2nl 2nl To sum up, Y4 can be (U , nk )-covered by no more than     nr nr + 1 + 2nk (r − k) + 1  2(nr + nk )(r − k + 1) nk nk many (U , nk )-names. This implies that for all sufficiently large r we have (n /n )−1

N (U0 r k , σ nk )  5(nr + 1) + 1 + 2nr (r − k) + 4(nr + nk )(r − k + 1)  n2r , which means that htop (U , σ ) = 0 and thus (y 1 , y 2 , y 3 ) ∈ / E3 (X, σ ). Now we modify (X, σ ) to get an invertible example. Let (X1 , T1 ) be the natural extension of (X, σ ). Then T1 is a homeomorphism and (X1 , T1 ) is transitive. Let Ai = π −1 (y i ), i = 1, 2, where π : X1 → X is the projection to the first coordinate. Then T1 (A1 ) ⊂ A2 and T1 (A 2 ) ⊂ A1 . It is not hard to check that T1 (B1 ) = B2 and T1 (B2 ) = B1 , where Bi = n1 T12n (Ai ), i = 1, 2. We claim that if K is a maximal entropy set with non-empty intersections with B1 and B2 then it is contained in B1 ∪ B2 . In fact, if there is x ∈ K\(B1 ∪B2 ), then (x, x1 , x2 ) is a 3-tuple, where xi ∈ K ∩Bi . It is clear that T1n (x) ∈ A1 ∪ A2 for each n ∈ Z (otherwise (π(T1n (x)), y 1 , y 2 ) is an entropy tuple). On the other hand, as x ∈ B1 ∪ B2 we have that there is n
1 with x ∈ T1n−1 (A1 ∪ A2 )\T1n (A1 ∪ A2 ), which implies that T1−n x ∈ T1−1 (A1 ∪ A2 )\(A1 ∪ A2 ), a contradiction, and hence our claim holds. Let (Y, S) be the TDS obtained by shrinking Bi to a point pi (i = 1, 2) with p1 = p2 . It is easy to check that (Y, S) is transitive, invertible and {p1 , p2 } is a maximal entropy set of (Y, S) by using theorem 2.10.  There are some open questions: e Question  1. Let (X, T ) be a TDS, µ ∈ M (X, T ) and hµ (T ) > 0. Assume that µ = Y µy dν(y) is the disintegration of µ over the Pinsker factor (Y, D, ν, S). Is it true that for ν-almost every y ∈ Y , supp(µy ) is a maximal entropy set for µ?

Question 2. Let (X, T ) be a TDS with positive entropy. Is any finite maximal entropy set a periodic orbit? Acknowledgments The first author is partially supported by NNSF of China (no. 10401031) and the second author is supported by the national key project for basic science (973). We remark that after finishing a preliminary version of the paper, Huang informed us that he had some results related to the topic. We thank him for the discussion. We would also like to thank the referees of the paper for their careful reading and useful comments which greatly improved the writing of the paper.

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