arXiv subjects

Vincent Béhani

Publications and source records attributed to Vincent Béhani.

2 recordsLinked to original sources

arXiv2023

Linear dynamics of an operator associated to the Collatz map

In this paper, we study the dynamics of an operator $\mathcal T$ naturally associated to the so-called Collatz map, which maps an integer $n \geq 0$ to $n / 2$ if $n$ is even and $3n + 1$ if $n$ is odd. This operator $\mathcal T$ is defined on certain weighted Bergman spaces $\mathcal B ^ 2 _ ω$ of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that $\mathcal T$ is hypercyclic on $\mathcal B ^ 2 _ ω$, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight $ω$, we show that $\mathcal T$ is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic.

math.FA↗

arXiv2022

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

In this paper, we study the so-called Bishop operators $T _ α$ on $L ^ p ([0, 1])$, with $α\in (0, 1)$ 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 $T _ α$. Building on results by Chalendar and Partington in the case where $α$ is rational, we show that $T _ α$ is cyclic for a dense $G _ δ$-set of irrational $α$'s, discuss cyclic functions and provide conditions in terms of convergents of $α\in \mathbf R \backslash \mathbf Q$ implying that certain functions are cyclic.

math.FA↗