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 $n\ge 0$ to $n\mathrm{/}2$ if $n$ is even and $3n+1$ if $n$ is odd. This operator $\mathcal{T}$ is defined on certain weighted Bergman spaces ${\mathcal{B}}_{\omega }^2$ of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that $\mathcal{T}$ is hypercyclic on ${\mathcal{B}}_{\omega }^2$, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight $\omega$, we show that $\mathcal{T}$ is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic.