Collatz map as a non-singular transformation
Volume 275 / 2024
Abstract
Let $T$ be the map defined on $\mathbb{N} =\{1,2,\ldots \}$ by $T(n) = {n}/{2} $ if $n$ is even and $T(n) = (3n+1)/2$ if $n$ is odd. Consider the dynamical system $(\mathbb {N}, 2^{\mathbb {N}}, T,\mu )$ where $\mu $ is the counting measure. This dynamical system has the following properties:
1. There exists an invariant finite measure $\gamma $ such that $\gamma (A) \leq \mu (A) $ for all $A \subset \mathbb{N}$.
2. For each function $f\in L^1(\mu )$ the averages $\frac {1}{N} \sum _{n=1}^N f(T^nx)$ converge for every $x\in \mathbb{N} $ to $ f^*(x)$ where $ f^* \in L^1(\mu )$.
We also show that the Collatz conjecture is equivalent to the existence of a finite measure $\nu $ on $(\mathbb{N} , 2^{\mathbb{N} })$ making the operator $Vf = f\circ T$ power bounded in $L^1(\nu )$ with conservative part $\{1,2\}$.
