  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 $\mathcal B ^ 2 _ \omega$ of
analytic functions on the unit disk. Building on previous work of Neklyudov, we
show that $\mathcal T$ is hypercyclic on $\mathcal B ^ 2 _ \omega$,
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.
