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

TL;DR

The paper investigates distributional chaos for composition operators $T_{\varphi}$ on $L^{p}$ spaces, establishing characterizations and practical criteria in terms of the inducing map $\varphi$ and measurable sets. It proves that distributional chaos on $L^{p}$ is equivalent to distributional chaos on $L^{p'}$ for all $p,p'$, and it provides a readily verifiable sufficient condition. It further shows that a dense set of distributionally irregular vectors implies a dense distributionally chaotic set, and it develops dense-chaos criteria. The work then specializes to shifts on weighted $\ell^{p}$ spaces, showing equivalences between distributional chaos and dense distributional chaos for bilateral/unilateral backward shifts and forward shifts under appropriate weight sequences. Finally, for composition operators on $L^{p}(\mathbb{T})$ induced by disk automorphisms, the operator is densely distributionally chaotic precisely when the automorphism has no fixed point in the disk, unifying Li-Yorke and distributional chaos with topological transitivity properties via no-fixed-point dynamics.

Abstract

In this paper, we investigate the distributional chaos of the composition operator $T_{\varphi}:f\mapsto f\circ\varphi$ on $L^{p}(X,\mathcal{B},μ)$, $1\leq p <\infty$. 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 $L^{p}(\mathbb{T},\mathcal{B},λ)$ 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 $\mathbb{D}$.

Theorems & Definitions (36)

• Definition 2.1: Distributional Chaos Criterion (DCC)
• Lemma 2.2: Theorem 12 Bernardes2013
• Theorem 2.3
• proof : Proof of Theorem \ref{['dcsufandnec']}
• Corollary 2.4
• Proposition 2.5
• proof : Proof of Proposition \ref{['dcsuf']}
• Theorem 2.6: Theorem 15 Bernardes2013
• Theorem 2.7
• proof