Relative entropy dimension of topological dynamical systems

We introduce the notion of relative topological entropy dimension to classify the different intermediate levels of relative complexity for factor maps. By considering the dimension or ''density" of special class of sequences along which the entropy is encountered, we provide equivalent definitions of relative entropy dimension. As applications, we investigate the corresponding localization theory and obtain a disjointness theorem involving relative entropy dimension.

1. Introduction. Since entropy was introduced by Kolmogorov from information theory, it has played an important role in the study of dynamical systems. Shannon's entropy (information entropy) is known to be the average information content in information science. Entropy (resp. relative entropy) is an important conjugate invariant for a dynamical system (resp. a factor map). Since zero entropy systems make up a dense G δ subset of all homeomorphisms, there are several kinds of works about conjugate invariants of zero entropy systems, for example, sequence entropy [19,15], maximal pattern entropy [13], entropy dimension [2,6,3] and Slow entropy [17]. But there is no relative conjugate invariants for zero entropy factor maps. We are attempting to systematically study relative zero-entropy invariants, especially relative invariants that can classify the different intermediate levels of relative complexity. Entropy dimension for a topological dynamical system first introduced by M. De Carvalho in [2] 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. S. Ferenczi and K. K. Park have introduced the entropy dimension in [6] to measure the complexity of entropy zero measurable dynamics. It measures the growth rate of H( n−1 i=0 T −i P ). D. Dou, W. Huang and K. K. Park in [3] introduced the notion of the dimension (upper and lower) of a subset of Z with density 0. They used the dimension of a special class of sequences which were called entropy generating sequences to measure the complexity of a topological system and showed that the topological entropy dimension can be computed through the dimensions of entropy generating sequences. We would like to define these notions for the relative setting and study their properties.
In positive entropy systems, the "independence" point of view enables people to have a better understanding how the complexity of a system produces. One can 6632 XIAOMIN ZHOU find an infinite subset W ⊂ Z + in the case of positive entropy such that along the sequence W , the symbolic names are "independent" and there exists c > 0 with lim inf n→∞ |W ∩[1,n]| n > c. We prove that if a system (X, T ) relevant to a factor map has positive relative entropy dimension, then there exists an relative entropy generating sequence S ⊂ Z + which is a union of disjoint finite sets along which the dynamics are "independent". Given a sequence S ⊂ Z + , we define the dimension of the sequence and show that the relative entropy dimension of the system is the supremum of the dimensions of the relative entropy generating sequences.
H. Furstenberg in [7] first introduced the concept of disjointness to characterize the difference of dynamical behavior between two systems. Two well-known examples are that in measurable dynamics K−mixing systems are disjoint from ergodic zero entropy systems and that weak mixing systems are disjoint from group rotations. In the case of topological settings, these properties are explored in [1,10,12,11]. D. Dou, W. Huang and K. K. Park in [3] introduced the notion of the dimension set D(X, T ) ⊂ [0, 1] of a zero entropy topological system (X, T ) to measure the various levels of topological complexity of subexponential growth rate. They investigated the property of disjointness within entropy zero systems via the dimension set and proved that under some conditions on the dimension sets and minimality, two dynamical systems of disjoint dimension sets are disjoint. This is a refinement and also a generalization of the result that uniformly positive entropy(u.p.e.) systems are disjoint from minimal and entropy zero systems. Relevant results for measure-theoretic settings can be found in recent work by D. Dou, W. Huang and K. K. Park in [4]. In this paper we would like to introduce the notion of relative dimension tuples and the relative dimension set and prove that two extensions of disjoint relative dimension sets for all orders are disjoint over the same dynamic system under some conditions. This can also be regarded as a a generalization of the result that an open rel.-u.p.e. extension of all orders is disjoint from any relative minimal and relative zero entropy extension in [14].
The paper is organized as follows. In Sect. 2, we give the definitions and some basic properties of relative entropy dimension. In Sect. 3, we consider the dimensions of the relative entropy generating sequence and investigate the interrelations among these dimensions. In Sect. 4, we give the notions of relative dimension tuples and dimension sets and prove disjointness theorem with respect to the relative dimension set.

2.
Relative entropy dimension. In this paper, a topological dynamical system (TDS, for short) is a pair (X, T ), where X is a compact metric space endowed with a self-homeomorphism T . Before we introduce the notion of relative entropy dimension for a TDS, we recall some definitions. Given a TDS (X, T ), denote by C X the set of finite covers of X and C o X the set of finite open covers of X. Given two covers U, V ∈ C X , we say that U is finer than Clearly, U ∨ V U and U ∨ V V. Given integers m ≤ n and a cover U ∈ C X let U n m = n i=m T −i U. Let (X, T ) and (Y, S) be two TDSs. Suppose that (Y, S) is a factor of (X, T ) in the sense that there exists a continuous surjective map π : (X, T ) → (Y, S) such that π • T = S • π. The map π is called a factor map from X to Y .
Let π : (X, T ) → (Y, S) be a factor map and U ∈ C o X . For E ⊆ X, let N (U, E) denote the number of the sets in a subcover of U which covers E with smallest cardinality and define N (U|π) = sup y∈Y N (U, π −1 (y)). For α ≥ 0, we define It is clear that D(T, U, α|π) does not decrease as α decreases, and D(T, U, α|π) / ∈ {0, +∞} for at most one α ≥ 0. We define the relative upper entropy dimension of Similarly, D(T, U, α|π) does not decrease as α decreases, and D(T, U, α|π) / ∈ {0, +∞} for at most one α ≥ 0. We define the relative lower entropy dimension of U relevant to π by If D(T, U|π) = D(T, U|π) = α, then we say U has relative entropy dimension α. Clearly 0 ≤ D(T, U|π) ≤ D(T, U|π) ≤ 1 and if h(T, U|π) > 0, then the relative entropy dimension of U is equal to 1.
Definition 2.1. Let π : (X, T ) → (Y, S) be a factor map between TDSs. The relative upper (resp. lower) entropy dimension of TDS (X, T ) relevant to π is When D(T |π) = D(T |π), we just call the common value the relative entropy dimension of (X, T ) relevant to π, denoted by D(T |π). When (Y, S) is a trivial system, we recover the classical entropy dimension(see [3] ), and in this case we shall omit the restriction on π. The following two propositions are the basic properties of relative entropy dimension. Hence lim sup This implies D(T, U ∨ V|π) < α. Since α is arbitrary, we have By (1) we have the result.
For (4), from (1) the first inequality is obvious. If α > max{D(T, U|π), D(T, V|π), min{D(T, U|π), D(T, V|π)}, without loss of generality we assume α > D(T, U|π) and α > D(T, V|π). Then As a direct application of Proposition 2.2, we have Let (X, T ) be a TDS. A cover {U, V } of X which consists of two non-dense open sets of X is called a standard cover of X. Denote by C s X the set of all standard covers of X. The following proposition shows that the relative upper entropy dimension with respect to standard covers determine the relative upper entropy dimension of the system. Proposition 2.4. Let π : (X, T ) → (Y, S) be a factor map between TDSs. Then Proof. We follow the argument in the proof of Proposition 1 in [1]. In the following, we will investigate the dimension of a special kind of sequence, which we call the relative entropy generating sequence.
Let π : (X, T ) → (Y, S) be a factor map between TDSs and U ∈ C o X . We say an increasing sequence of integers S = {s 1 < s 2 < · · · } is a relative entropy generating Denote by E(T, U|π) the set of all relative entropy generating sequences of U relevant to π and by P(T, U|π) the set of sequence S = {s 1 < s 2 < · · · } of Z + with the property that In other words, P(T, U|π) is the set of increasing sequence of integers along which U has positive relative entropy.
Definition 3.1. Let π : (X, T ) → (Y, S) be a factor map between TDSs and U ∈ C o X . We define Similarly, we can define D e (T, U|π) and D p (T, U|π) by changing the upper dimension into the lower dimension. Similarly, we can define D e (T |π) and D p (T |π).
Let k ≥ 2 and B be a nonempty finite subset of Z + . Assume U is the cover of {0, 1, · · · , k} B = z∈B {0, 1, · · · , k} consisting of subsets of the form z∈B {i z } c , where 1 ≤ i z ≤ k and {i z } c = {0, 1, · · · , k} \ {i z } for each z ∈ B. For S ⊆ {0, 1, · · · , k} B we let C S denote the minimal cardinality of subcovers of U one needs to cover S. Note that we shall use natural logarithms unless we explicitly indicate otherwise.
The proof of this lemma is completely similar to the proof of Lemma 3.7 in [3]. With the help of above lemma, we have the following theorem. Let τ j−1 < η j < τ j for j ∈ N. By Lemma 3.4, there exists N j ∈ N such that for every finite set B with |B| ≥ N j and N ( i∈B T −i U|π) ≥ e a 2 |B| τ j , we can find W ⊆ B with |W | ≥ |B| ηj and {A 1 , A 2 } which is independent along W relevant to π.
This shows F ∈ E(T, U|π). Note that lim sup U|π). This finishes the proof.
If U c 1 is a singleton then we put U 1 1 = U 1 . If U c 1 has at least two different points y and y , fix 1 > 0 with 1 ≤ d(y,y ) 4 , and construct a cover of U c 1 by open balls with radius 1 centered in U c 1 ; call it A. Since U c 1 is compact, there exist A 1 , A 2 , · · · , A u ∈ A such that , · · · , u. By the choice of 1 , each closed set F i is a proper subset of U c 1 with diam(F i ) ≤ 1 2 diam(U c 1 ). Since By the above a), we have ∞ j=1 (U j 1 ) c = {x 1 }, · · · , ∞ j=1 (U j n ) c = {x n } for some x 1 , . . . , x n in X. Moreover, since {U 1 , . . . , U n } covers X and x i ∈ (U i ) c , we have (x i ) n 1 ∈ X (n) \ ∆ n (X). Finally, we show D(x 1 , . . . , x n |π) ≥ D(T, U|π). Since (x i ) n 1 ∈ X (n) \ ∆ n (X), without loss of generality, we assume x 1 = x n . There exists k 0 ∈ N such that B(x 1 , 1 k0 ) ∩ B(x n , 1 k0 ) = ∅. For any k ≥ k 0 , there exists j(k) ∈ N large enough such that (U This finishes the proof of the lemma. Proposition 4.3. Let π : (X, T ) → (Y, S) and π : (Z, R) → (X, T ) be two factor maps.