Relative entropy dimension of topological dynamical systems

Abstract

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.2010 Mathematics Subject Classification. Primary: 37B99, 54H20. Key words and phrases. Relative entropy dimension, relative dimension set. (*) This research is supported by NSFC(11801193). 6631 6632 XIAOMIN ZHOUfind 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

Relative entropy dimensions for amenable group actions

“…(Cf. [22] for G = Z) Let π : (X, G) → (Y, G) be a factor map between G-systems, and let A 1 , A 2 be two disjoint non-empty closed subsets of X,…”

“…Although systems with positive entropy are much more complicated than those with zero entropy, zero entropy systems own various levels of complexity, and recently have been discussed in [3,4,6,7,9,13,18,22]. Those authors adopted various methods to classify zero entropy dynamical systems.…”

“…And then they used the dimension of a special class of sequences which were called entropy generating sequences to measure the complexity of a system and showed that the topological entropy dimension can be computed through the dimensions of entropy generating sequences. Zhou [22] defined the corresponding notions for Z-actions and studied their properties. In this paper, we would like to redefine the relevant notions in relative setting for group actions through introducing a new notion which we call relative "relative entropy generating sets".…”

“…This is a refinement and also a generalization of the result in [2] that uniformly positive entropy systems are disjoint from minimal entropy zero systems. Based on the above results, Zhou [22] introduced the notion of relative dimension tuples and the relative dimension set and proved that two extensions with disjoint relative dimension sets for all orders are disjoint over the same system under some conditions. This can be also regarded as a generalization of the results in [16] that an open extension with relative uniformly positive entropy of all orders is disjoint from any minimal extension with relative zero entropy.…”