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

TL;DR

This paper investigates the power-boundedness of Syracuse and Collatz maps in measure-theoretic terms, establishing conditions for cycles and exploring inverse image properties with implications for the Collatz Conjecture.

Contribution

It characterizes power-boundedness of Syracuse maps and their inverses, linking dynamical properties to number theory and the Collatz Conjecture.

Findings

Power-boundedness in $L^1(\mu)$ is equivalent to the existence of a cycle.

Inverse images of the maps have specific structural properties.

The study relates inverse image density to the dynamics of the maps.

Abstract

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. We also analyze some structural properties of the inverse image in relation to asymptotic density of the set $\{x \in \mathbb{N} \mid \exists k \in \mathbb{N}: T^k(x) < x\}$.