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

In this paper, we study the so-called Bishop operators $T_{\alpha }$ on $L^p([0,1])$, with $\alpha \in (0,1)$ and $1<p<+\mathrm{\infty }$, 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 $\alpha \in \mathbf{R}\mathrm{\backslash }\mathbf{Q}$ implying that certain functions are cyclic.