On the mixing properties of piecewise expanding maps under composition with permutations

TL;DR

This paper investigates how composing a piecewise expanding map with permutations affects its mixing properties, showing most permutations preserve mixing but typically worsen the mixing rate, with detailed analysis for the stretch-and-fold map.

Contribution

It provides a combinatorial characterization of permutations that preserve mixing and analyzes how permutations influence the mixing rate of piecewise expanding maps.

Findings

Most permutations preserve topological mixing as N grows large.

Permutation composition generally worsens the mixing rate of the map.

The worst-case mixing rate approaches 1 as N increases, indicating slow mixing.

Abstract

We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \bmod 1$ for integers $m \geq 2$ is examined in detail. We give a combinatorial description of those permutations $\sigma$ for which $\sigma \circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \to \infty$. We then investigate the mixing rate of $\sigma \circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $\sigma$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N \to \infty$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.