On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

TL;DR

This paper studies how composing piecewise linear maps with permutations affects their mixing properties, extending previous work to include maps with non-constant orientation.

Contribution

It generalizes earlier results by analyzing the impact of permutations on a broader class of piecewise linear maps with variable orientation.

Findings

Permutation composition can enhance mixing in certain cases.

The effect depends on the specific permutation and map structure.

Extensions to non-constant orientation maps are established.

Abstract

For an integer $m \geq 2$, let $\mathcal{P}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let $\mathcal{F}_m$ be the class of piecewise linear maps on $I$ with constant slope $\pm m$ on each element of $\mathcal{P}_m$. We investigate the effect on mixing properties when $f \in \mathcal{F}_m$ is composed with the interval exchange map given by a permutation $\sigma \in S_N$ interchanging the $N$ subintervals of $\mathcal{P}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\bf 33}, (2013) 3365--3390], where we considered only the "stretch-and-fold" map $f_{sf}(x)=mx \bmod 1$.