2023/03/06 by Béhani, Vincent
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2303.03203
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 ≥ 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.