Entropy for A-coupled-expanding Maps and Chaos

TL;DR

This paper investigates the lower bounds of topological entropy for A-coupled-expanding maps, establishing criteria for chaos, and provides entropy estimates for specific maps including circle maps and the Kasner map.

Contribution

It introduces generalized lower bounds for topological entropy in A-coupled-expanding maps and explores conditions for these maps to be factors of subshifts of finite type.

Findings

Lower bounds of topological entropy are established for A-coupled-expanding maps.

Conditions are derived for maps to be factors of subshifts of finite type.

Topological entropy for circle maps and the Kasner map is explicitly calculated.

Abstract

The concept of "$A$-coupled-expanding" map for a transition matrix $A$ has been studied as one of the most important criteria of chaos in the past years. In this paper, the lower bound of the topological entropy for strictly $A$-coupled-expanding maps is studied as a criterion for chaos in the sense of Li-Yorke, which is less conservative and more generalized than the latest result is presented. Furthermore, some conditions for $A$-coupled-expanding maps excluding the strictness to be factors of subshifts of finite type are derived. In addition, the topological entropy of partition-$A$-coupled-expanding map, which is put forward in this paper, is further estimated on compact metric spaces. Particularly, the topological entropy for partition-$A$-coupled-expanding circle maps is given, with that for the Kasner map being calculated for illustration and verification.