Syracuse Maps as Non-singular Power-Bounded Transformations and Their Inverse Maps

Idris Assani, Ethan Ebbighausen, Anand Hande

Published 2022-08-25Version 1

We prove that the dynamical system $(\mathbb{N}, 2^{\mathbb{N}}, T, \mu)$, where $\mu$ is a finite measure equivalent to the counting measure, is power-bounded in $L^1(\mu)$ if and only if there exists one cycle of the map $T$ and for any $x \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that $T^k(x)$ is in some cycle of the map $T$. This result has immediate implications for the Collatz Conjecture, and we use it to motivate the study of number theoretic properties of the inverse image $T^{-1}(x)$ for $x \in \mathbb{N}$, where $T$ denotes the Collatz map here. We study similar properties for the related Syracuse maps, comparing them to the Collatz map.