A study of Bishop operators from the point of view of linear dynamics

TL;DR

This paper analyzes Bishop operators on L^p spaces from a linear dynamics perspective, showing they are not hypercyclic or supercyclic, and exploring their cyclicity properties, especially for irrational parameters.

Contribution

It provides new insights into the cyclicity and dynamic properties of Bishop operators, extending previous rational cases to irrational parameters.

Findings

Bishop operators are never hypercyclic or supercyclic.

They are cyclic for a dense G_delta set of irrational alpha.

Conditions for cyclicity depend on convergents of alpha.

Abstract

In this paper, we study the so-called Bishop operators $T _ \alpha$ on $L ^ p ([0, 1])$, with $\alpha \in (0, 1)$ and $1 < p < + \infty$, from the point of view of linear dynamics. We show that they are never hypercyclic nor supercyclic, and investigate extensions of these results to the case of weighted translation operators. We then investigate the cyclicity of the Bishop operators $T _ \alpha$. Building on results by Chalendar and Partington in the case where $\alpha$ is rational, we show that $T _ \alpha$ is cyclic for a dense $G _ \delta$-set of irrational $\alpha$'s, discuss cyclic functions and provide conditions in terms of convergents of $\alpha \in \mathbf R \backslash \mathbf Q$ implying that certain functions are cyclic.