Topological transivity and mixing of the composition operators

TL;DR

This paper characterizes when composition operators on $L^p$ spaces are topologically transitive or mixing, based on properties of the underlying transformation, and extends results to other Banach spaces.

Contribution

It provides necessary and sufficient conditions for topological transitivity and mixing of composition operators, including those induced by specific transformations like weighted shifts and odometers.

Findings

Characterizes topological transitivity of composition operators.

Provides criteria for topological mixing.

Extends results to general Banach spaces.

Abstract

Let $X=(X,\mathcal{B},\mu)$ be a $\sigma$-finite measure space and \mbox{$f:X\to X$} be a measurable transformation such that the composition operator $T_f:\varphi\mapsto \varphi\circ f$ is a bounded linear operator acting on $L^p(X,\mathcal{B},\mu)$, $1\le p<\infty$. We provide a necessary and sufficient condition on $f$ for $T_f$ to be topologically transitive or topologically mixing. We also characterize the topological dynamics of composition operators induced by weighted shifts, non-singular odometers and inner functions. The results provided in this article hold for composition operators acting on more general Banach spaces of functions.