Collatz map as a non-singular transformation

Abstract

Let $T$ be the map defined on $\mathbb{N}=\{1,2,3,...\}$ by $T(n)=\frac{n}{2}$ if $n$ is even and by $T(n)=\frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\mathbb{N},2^{\mathbb{N}},T,\mu )$ where $\mu$ is the counting measure. This dynamical system $(\mathbb{N},2^{\mathbb{N}},T,\mu )$ has the following properties. \begin{enumerate} \item There exists an invariant finite measure $\gamma$ such that $\gamma (A)\le \mu (A)$ for all $A\subset \mathbb{N}.$ \item For each function $f\in L^1(\mu )$ the averages $\frac{1}{N}{\sum }_{n=1}^{N}f(T^{n}x)$ converge for every $x\in \mathbb{N}$ to $f^{\ast }(x)$ where $f^{\ast }\in L^1(\mu ).$ \end{enumerate} 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 conserrvative part $\{1,2\}.$

This is the final version of the paper accepted for publication (in Studia Math.)