Distributional chaos for composition operators on $L^{p}$-spaces

Abstract

In this paper, we investigate the distributional chaos of the composition operator $T_{\varphi}:f\mapsto f\circ\varphi$ on $L^{p}(X,\mathcal{B},\mu)$, $1\leq p <\infty$. We provide a characterization and practical sufficient conditions on $\varphi$ for $T_{\varphi}$ to be distributionally chaotic. Furthermore, we show that the existence of a dense set of distributionally irregular vectors implies the existence of a dense distributionally chaotic set, without any additional condition. We also provide a useful criterion for densely distributional chaos. Moreover, we characterize the weight sequences that ensure distributional chaos for bilateral backward shifts, unilateral backward shifts, bilateral forward shifts, and unilateral forward shifts on the weighted $\ell^{p}$-spaces $\ell^{p}(\mathbb{N},v)$ and $\ell^{p}(\mathbb{Z},v)$. As a consequence, we reveal the equivalence between distributional chaos and densely distributional chaos for backward shifts and forward shifts on $\ell^{p}(\mathbb{Z},v)$ without any additional condition. Finally, we characterize the composition operator $T_{\varphi}$ on $L^{p}(\mathbb{T},\mathcal{B},\lambda)$ induced by an automorphism $\varphi$ of the unit disk $\mathbb{D}$. We show that $T_{\varphi}$ is densely distributionally chaotic if and only if $\varphi$ has no fixed point in $\mathbb{D}$.