Linear dynamics of an operator associated to the Collatz map

Abstract

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