Hypercyclic Composition Operators]{Hypercyclic composition operators on the little Bloch space $\mathcal{B}_0$ and the Besov spaces $B_p

TL;DR

This paper proves that no composition operators induced by holomorphic self-maps of the unit disk are hypercyclic on the little Bloch space and Besov spaces, clarifying the limitations of operator dynamics in these function spaces.

Contribution

The paper establishes the non-existence of hypercyclic composition operators on the little Bloch and Besov spaces, providing new insights into operator behavior in these spaces.

Findings

No hypercyclic composition operators on $\

ext{little Bloch space } \\mathcal{B}_0$ and Besov spaces $B_p$.

Abstract

Let $S(\mathbb{D})$ be the collection of all holomorphic self-maps on $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_{\varphi}$ the composition operator induced by $\varphi\in S(\mathbb{D})$. We obtain that there are no hypercyclic composition operators on the little Bloch space $\mathcal{B}_0$ and the Besov space $B_p$.