ZERO SEQUENCE ENTROPY AND ENTROPY DIMENSION

. Let ( X,T ) be a topological dynamical system and M ( X ) the set of all Borel probability measures on X endowed with the weak ∗ -topology. In this paper, it is shown that for a given sequence S , a homeomorphism T of X has zero topological sequence entropy if and only if so does the induced homeomorphism T of M ( X ). This extends the result of Glasner and Weiss [9, Theorem A] for topological entropy and also the result of Kerr and Li [15, Theorem 5.10] for null systems. Moreover, it turns out that the upper entropy dimension of ( X,T ) is equal to that of ( M ( X ) ,T ). We also obtain the version of ergodic measure-preserving systems related to the sequence entropy and the upper entropy dimension.

1. Introduction. Throughout this paper, by a topological dynamical system (TDS, for short) we mean a pair (X, T ), where X is a compact metric space with the metric d and T is a homeomorphism from X to itself. We denote by M (X) the collection of all Borel probability measures on X endowed with the weak * -topology and by M (X, T ) the collection of all T -invariant Borel probability measures on X.
For a TDS (X, T ), Adler, Konheim and McAndrew [1] introduced the notion of topological entropy as an invariant of topological conjugacy in 1965. Before this, Kolmogorov defined the measure-theoretical entropy in 1958. For µ ∈ M (X, T ), Kushnirenko [16] extended this notion to the measure-theoretical sequence entropy, which is also an invariant, i.e. h S µ (T ) with respect to an increasing positive sequence S, and showed that a measurable system has zero sequence entropy for any sequence if and only if the system has discrete spectrum. Then, Goodman [11] defined the topological sequence entropy h S top (T ) with respect to an increasing positive sequence S. For a minimal system, a similar version of Kushnirenko's result holds, that is, if a minimal system has zero topological sequence entropy for any sequence then it is an almost 1-1 extension of an equicontinuous system (see [12]); moreover, for a subshift which is null, the extension is 1-1 on a set of haar measure one (see [7]).
In addition, one can also formulate the properties of mixing in terms of sequence entropy. For further study on these topics see e.g., [14,18,9,13,15].
A TDS (X, T ) can induce a system (M (X), T ) in a natural way. Bauer and Sigmund [2] first systematically investigated the connection of dynamical properties between (X, T ) and (M (X), T ), which has now been studied more and more extensively. Glasner and Weiss [9, Theorem A] showed a classical result that the system (X, T ) has zero topological entropy if and only if so does the induced system (M (X), T ). This method attracts a lot of attention since one of the proofs established a remarkable connection between the entropy theory and a well-known combinatorial lemma due to Sauer, Pereles and Shelah. Later, this connection was further developed by Kerr and Li [15], in which a similar result, that (X, T ) is null if and only if (M (X), T ) is null (recall that a TDS (X, T ) is null if the topological sequence entropy of (X, T ) is zero for any increasing positive sequence), was obtained.
In our current paper, we firstly investigate the connection of topological sequence entropies between (X, T ) and (M (X), T ). Using the idea in [9], we strengthen [9, Theorem A]. More specifically, we will provide a more direct proof without the use of zero dimensional extensions to show that this result also holds for the topological sequence entropy.
There are different ways to measure the complexity of a topological system, for example topological entropy, topological sequence entropy and entropy dimension. Entropy dimension D(X, T ) of a t.d.s. (X, T ) systematically studied by Dou-Huang-Park [5] measures the superpolynomial but subexponential growth rate of the number of open sets that cover the space out of the sequence of iterated open covers. Particularly, it only works on topological systems with zero entropy. As an analogy of topological entropy dimension, the measure-theoretical version of the entropy dimension D µ (X, T ) for µ ∈ M (X, T ) was also defined to measure the growth rate of iterated partitions in [4]. It should be mentioned unfortunately that, there is no variational principle between the topological entropy dimension and the measure-theoretical entropy dimension (see [5, example 4.6]).
In this paper, we are interested in the investigation how the complexity of (X, T ) relates to the complexity of (M (X), T ). It is already known that h top (X, T ) = 0 if and only if h top (M (X), T ) = 0. As we have said in the previous paragraph, our first aim is to get a similar result for topological sequence entropy. Then we study the relationship of entropy dimension between (X, T ) and (M (X), T ). The topological entropy dimension can be characterized by the dimension of increasing sequences along which the topological sequence entropies are positive (see [5]). This makes it possible for us to research entropy dimension using sequence entropy. Therefore as an application of the result we extend to the case of the topological sequence entropy, we will prove that the upper entropy dimension of (X, T ) is equal to that of (M (X), T ). We state them together as follows. For measure-preserving systems, it is expected to obtain the similar results as the above theorem. In order to do this, the notion of quasi-factors is needed (see [10]), which is a natural generalization of factors. It was shown that, like factors, quasi-factors inherit some dynamical properties. For example, zero entropy and distality are preserved by quasi-factors (see [9]). For µ ∈ M (X, T ), a quasi-factor k of µ is any T -invariant measure on M (X) whose barycenter is µ; that is, We now state our result of the measure-preserving version in the following. This paper is organized as follows. In Section 2, we introduce some definitions and related results on the sequence entropy and the entropy dimension. In Section 3, we give the proof of Theorem 1. In Section 4, we state some useful results about the notion of quasi-factors and joinings, and finally prove Theorem 2.
2. Preliminaries. In this section, we firstly give some basic notions and fundamental properties. For a TDS (X, T ), denote by B(X) the collection of all Borel subsets of X, C X the set of all finite covers of X, and C o X the set of all finite open covers of X. Given U, V ∈ C X , let U ∨ V = {U ∩ V : U ∈ U, V ∈ V} and U n m = ∨ n i=m T −i U, where m, n ∈ Z with m ≤ n. We can similarly define these notations for finite partitions of X. Given U ∈ C o X , let N (U) denote the number of the sets in a subcover of U with the smallest cardinality.
2.1. Topological sequence entropy. Let (X, T ) be a TDS, U ∈ C o X , and S = {s 1 < s 2 < · · · } be an increasing sequence of non-negative integers. The topological sequence entropy of T with respect to U and S is given by Sometimes, we write h S top (X, T, U) to emphasize the space X if necessary. The topological sequence entropy of T with respect to S is given by If S = Z + , we recover the standard topological entropy. Like topological entropy, there is an equivalent definition of topological sequence entropy by using separated sets and spanning sets along with increasing sequences (see [11]). For n ≥ 1 and > 0, we say that a finite set A ⊂ X is an (S, n, )-separated set if for any x, y ∈ A, x = y, there exists i ∈ {1, 2, · · · n} such that d(T si x, T si y) ≥ . Denote by s(S, n, ) the largest cardinality of any (S, n, )-separated set. We now define s(S, ) = lim sup Note that this limit exists since 1 < 2 implies s(S, 1 ) ≥ s(S, 2 ). It is well known that h S top (T ) = s(S, d). It turns out from the following result (see [11]) that topological sequence entropy is an isomorphic invariant.
Lemma 2.1. If X 1 and X 2 are compact metric spaces, T i : X i → X i are continuous for i = 1, 2, and if φ : Measure-theoretic sequence entropy. For a TDS (X, T ), let α be a finite partition of X, µ ∈ M (X, T ), and S = {s 1 < s 2 < s 3 · · · } an increasing sequence of non-negative integers. Define the measure theoretic sequence entropy of T with respect to α along S by where the supremum is taken over all finite measurable partitions of X. Similarly, the case S = Z + recovers the standard measure-theoretic entropy.
Note that if C = {C i : i = 1, 2, · · · n} is a finite sub-σ-algebra of B(X), then the non-empty sets B 1 ∩ B 2 · · · ∩ B n , where B i = C i or X \ C i , form a finite partition of B(X), which is denoted by ξ(C). Conversely, if ζ = {A 1 , A 2 , · · · , A k } is a finite partition of X, then the set consisting of all those elements which are unions of some A i 's is a finite sub-σ-algebra of B(X). We denote it by A(ζ). It is not hard to check that A(ξ(C)) = C and ξ(A(ζ)) = ζ. So there is no confusion to define The following result is taken from [16].
and S = {s 1 < s 2 < · · · } an increasing sequence of non-negative integers. Then the following hold: A n denotes the smallest σ-algebra containing all A n for n ≥ 1. Suppose (X 1 , B(X 1 ), m 1 , T 1 ) and (X 2 , B(X 2 ), m 2 , T 2 ) are two measure-preserving systems. We say that T 2 is a factor of T 1 if there exists a measure-preserving transformation φ : X 1 → X 2 such that for m 1 -a.e. x ∈ X 1 . If in addition, φ is invertible and φ −1 is also measure-preserving, then we say that T 1 is isomorphic to T 2 . One can find the following lemma in [16].
for any increasing sequence S = {s 1 < s 2 < s 3 · · · } of non-negative integers. In particular, measuretheoretic sequence entropy is an isomorphic invariant. Similarly, we can define the lower dimension of S as Let (X, T ) be a TDS and U ∈ C o X . We denote by P(T, U) the set of all increasing sequences S = {s 1 < s 2 < · · · } of non-negative integers with the property that lim sup In other words, P(T, U) is the set of all increasing sequences of non-negative integers along which U has positive sequence entropy. We now define

Entropy dimension of topological dynamical systems.
Before giving a precise definition of entropy dimension, we firstly explain how it measures complexity. Entropy dimension intends to estimate the speed of N ( not only with n, as usually to calculate the topological entropy, but also with other powers of n. Roughly speaking, it corresponds to topological entropies at different speeds, taking advantage from the canonical relationship between the functions log(x) and x s for any s > 0. From the definition below, it is easy to check that the entropy dimension will be one when the system has positive entropy or infinite entropy and it will be zero for the null system. But a system with zero entropy dimension dose not mean that it is null, such as the Morse system whose complexity function has linear growth rate.
Let U ∈ C o X and a ≥ 0. Define

Remark 1.
It is easy to observe that for any positive entropy system, its entropy dimension exists and equals one and by Lemma 2.4, the entropy dimension of null system also exists and equals zero. Like topological entropy, the calculation of entropy dimension is also a tough work. Readers can refer to [6, Proposition 5] for the system with the existence of entropy dimension and [6, Proposition 4] for the system whose upper and lower entropy dimension are not the same. More examples whose upper entropy dimension have been computed are available in [3].
The following result characterizes the upper entropy dimension by the increasing sequences of integers along which the sequence entropy is positive.

2.3.2.
Measure-theoretic entropy dimension. Let µ ∈ M (X, T ) and α ∈ P X , where P X denotes the collection of all finite measurable partitions of X. We say that an increasing sequence S = {s 1 < s 2 < · · · } of non-negative integers is an entropy generating sequence of α if lim inf We say an increasing sequence S = {s 1 < s 2 < · · · } of non-negative integers is a positive entropy sequence of α if the sequence entropy of α along the sequence S is positive.
We define the upper entropy dimension of (X, B, µ, T ) by and the lower entropy dimension of (X, B, µ, T ) by If D µ (X, T ) = D µ (X, T ), then we define the entropy dimension D µ (X, T ) of (X, B, µ, T ) as the common value.
3. Proof of Theorem 1. We endow l n 1 = R n with the norm ||x|| = n i=1 |x i | and l n ∞ = R n with the norm ||x|| ∞ = max{|x i | : 1 ≤ i ≤ n}. The following lemma plays an important role in our proof of Theorem 1. Since X is a compact metric space, we can find a sequence f 1 , f 2 , · · · in C(X) such that f n ≤ 1 and {f n } ∞ n=1 is dense in the unit sphere around 0 in C(X). Then is a metric on M (X) giving the weak * -topology (see e.g., [17,Theorem 6.4], [19,Theorem 6.4] for details).
for any µ, ν ∈ E with µ = ν. Choose δ > 0 such that whenever d(x, y) < δ, To complete the proof of (1) of this theorem, it suffices to show that h S top (X, T, U) > 0. To do this, we divide it into three steps in the following.
Step 1. We will construct a linear mapping φ from l L N i 1 to l Ni ∞ with ||φ|| ≤ 1 for any i large enough.
By the definition of L Ni , we can take a subcover V = {V 1 , V 2 , · · · , V L N i } of Ni j=1 T −sj U with the minimal cardinality. Construct a partition of X as follows: Since A k = ∅ for any k = 1, 2, · · · , L Ni , we can take z k ∈ A k for any k = 1, 2, · · · , L Ni . Consider φ : l It is easy to observe that φ is a linear mapping from l Step 2. For such φ, we will prove that φ(B 1 (l L N i 1 )) contains more than e bNi points that are 0 9 -separated. Consider ψ : E → l L N i 1 , µ → (µ(A 1 ), µ(A 2 ), · · · , µ(A L N i )). It is clear that ) and φ(ψ(E)) ⊂ φ(B 1 (l L N i 1 )). Therefore, it remains to check that φ(ψ(E)) contains more than e bNi points that are 0 9 -separated.
Firstly we show that φ(ψ(E)) is 0 9 -separated. Suppose this is false, then there exist µ, ν ∈ E with µ = ν, such that On the other hand, where and Since z k ∈ A k ⊂ V k ∈ Ni j=1 T −sj U, we have that, for x ∈ A k and t = 1, 2, · · · , N i , there exists j ∈ {1, 2, · · · , d} such that T st x, T st z k ∈ U j . By the definition of U and V, we have d(T st x, T st z k ) < δ, and hence for any x ∈ A k and n = 1, 2, · · · , K. Thus, we have By the similar argument, we have that I 2 ≤ 0 9 . Note that ( * * ) means I 3 ≤ 0 9 . Therefore, for any 1 ≤ t ≤ N i . This is a contradiction with ( * ). Thus, φ(ψ(E)) are 0 9separated.

YIXIAO QIAO AND XIAOYAO ZHOU
Moreover, by noting that E contains more than e bNi points and that φ • ψ is injective on E, we obtain that φ(ψ(E)) contains more than e bNi points that are 0 9 -separated.
Step 3. We now show h S top (X, T, U) > 0 as follows. By Step 1 and Step 2, we know that when i is large enough, φ is a linear mapping from l )) contains more than e bNi points that are 0 9 -separated. Then by Lemma 3.1, there is n 0 ∈ N and a constant c > 0 such that for all N i > n 0 , we have L Ni ≥ 2 cNi . Thus, This ends the proof of (1).
(2) By Lemma 2.4, it suffices to show that D p (X, T ) = D p (M (X), T ). Put a := D p (X, T ). We have two cases in the following.   Remark 2. Now we generalize (1) of Theorem 1 to group actions. Let (X, G) be G-system, where X is a compact metric space with a metric d and G is a countable topological group. Given a sequence S = {g 1 , g 2 , g 3 , · · · } of G, we define the topological sequence entropy of (X, G) with respect to a finite open cover U of X and the sequence S by and the topological sequence entropy of (X, G) with respect to S by Similarly, we can define the topological sequence entropy by (S, n, )-separated sets. For n ≥ 1 and > 0, we say that a finite set A ⊂ X is an (S, n, )-separated set if for any x, y ∈ A with x = y, there exists i ∈ {1, 2, · · · n} such that d(g i x, g i y) ≥ . Denote by sr G (S, n, ) the largest cardinality of any (S, n, )-separated sets. Put It is not hard to check that h S top (X, G) = sr G (S, d). By the same argument as in the proof of (1) of Theorem 1, we can show the following result. Let (X, G) be a G-system and S = {g 1 , g 2 , g 3 , · · · } a sequence of G. For k ∈ Q(µ), define It is easy to prove that k ∞ ∈ J(µ, Z) and it is a bi-infinite self-joining. The following result characterizes the relationship between quasi-factors and their biinfinite selfjoinings. Then (M (X), B(M (X)), k, T ) is a factor of (X Z , B(X) Z , k ∞ , T Z ).
With the above preparation, we begin to prove Theorem 2.
Proof of Theorem 2. It is easy to verify that η is T -invariant and (M (X), B(M (X)), η, T ) is isomorphic to (X, B(X), µ, T ). Thus, by Lemma 2.3, we have that h S η (M (X), T ) = h S µ (X, T ). To prove η ∈ Q(µ), we need to show that µ is the barycenter of η. For a given f ∈ C(X), we have This shows that η ∈ Q(µ) which implies h S η (M (X), T ) = 0 by our assumption. Thus, h S µ (X, T ) = 0. To prove the converse, assume h S µ (X, T ) = 0. Firstly, we can choose an increasing is a countable topological basis of (X, B(X)). Let A n be the σ-algebra generated by {A i : i = 1, 2, · · · , n}. Then {A n } n≥1 is as required.