Collatz map as a non-singular transformation

Abstract

Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ 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 $(\N, 2^{\N}, T,\mu)$ where $\mu$ is the counting measure. This dynamical system $(\N, 2^{\N}, T, \mu)$ has the following properties. \begin{enumerate} \item There exists an invariant finite measure $\gamma$ such that $\gamma(A) \leq \mu(A) $ for all $A \subset \N.$ \item For each function $f\in L^1(\mu)$ the averages $\frac{1}{N} \sum_{n=1}^N f(T^nx)$ converge for every $x\in \N$ to $ f^*(x)$ where $ f^* \in L^1(\mu).$ \end{enumerate} We also show that the Collatz conjecture is equivalent to the existence of a finite measure $\nu$ on $(\N, 2^{\N})$ making the operator $Vf = f\circ T$ power bounded in $L^1(\nu)$ with conserrvative part $\{1,2\}.$