Linear dynamics of an operator associated to the Collatz map

Abstract

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.