Some properties of circle maps with zero topological entropy

TL;DR

This paper investigates properties of circle maps with zero topological entropy, establishing equivalences among non-separable, IN, and IT pairs, and analyzing pattern entropy growth for null maps.

Contribution

It characterizes non-separable pairs as IN and IT pairs and analyzes pattern entropy growth in topologically null circle maps.

Findings

Non-diagonal pairs are non-separable iff they are IN and IT pairs.

Topological null maps have polynomial order maximal pattern entropy.

Provides a characterization of pairs and entropy growth in zero entropy circle maps.

Abstract

For a circle map $f\colon\mathbb{S}\to\mathbb{S}$ with zero topological entropy, we show that a non-diagonal pair $\langle x,y\rangle\in \mathbb{S}\times \mathbb{S}$ is non-separable if and only if it is an IN-pair if and only if it is an IT-pair. We also show that if a circle map is topological null then the maximal pattern entropy of every open cover is of polynomial order.