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Topological transivity and mixing of the composition operator on Lp spaces

Research paper by Udayan B. Darji, Benito Pires

Indexed on: 04 Oct '16Published on: 04 Oct '16Published in: arXiv - Mathematics - Dynamical Systems



Abstract

Let $X=(X,\Sigma,\mu)$ be a $\sigma$-finite measure space and $f:X\to X$ be an one-to-one bimeasurable transformation satisfying $\mu\left(f(B)\right)\ge c_1 \mu(B)$ for some constant $c_1>0$ and every measurable set $B$. Then, $T_f:\varphi\mapsto \varphi\circ f$ is a bounded linear operator acting on $L^p(X,\Sigma,\mu)$, $1\le p<\infty$, called the composition operator induced by $f$. We provide necessary and sufficient conditions on $f$ for $T_f$ to be topologically transitive or topologically mixing. We also give two examples of one-to-one bimeasurable transformations whose composition operators are topologically transitive but not topologically mixing. Finally, we show that the composition operator induced by a bi-Lipschitz $\mu$-contraction (or more generally, by a $\mu$-dissipative transformation) defined on a finite measure space is always topologically mixing.