Chaos and Entropy for Interval Maps

TL;DR

This paper explores chaotic properties of interval maps, establishing relationships between entropy, Li-Yorke chaos, and nonseparable pairs, and characterizes conditions for positive entropy and pattern entropy in these systems.

Contribution

It provides new characterizations of chaos and entropy for zero and positive entropy interval maps, including the structure of nonseparable pairs and polynomial bounds on pattern entropy.

Findings

Proximal relation is an equivalence relation for zero entropy maps.

Sequence entropy pairs coincide with nonseparable pairs in zero entropy maps.

Interval maps with topological nullity have polynomial pattern entropy.

Abstract

In this paper, various chaotic properties and their relationships for interval maps are discussed. It is shown that the proximal relation is an equivalence relation for any zero entropy interval map. The structure of the set of $f$-nonseparable pairs is well demonstrated and so is its relationship to Li-Yorke chaos. For a zero entropy interval map, it is shown that a pair is a sequence entropy pair if and only if it is $f$-nonseparable. Moreover, some equivalent conditions of positive entropy which relate to the number "3" are obtained. It is shown that for an interval map if it is topological null, then the pattern entropy of every open cover is of polynomial order, answering a question by Huang and Ye when the space is the closed unit interval.