Collatz map as a power bounded nonsingular transformation

Idris Assani

Published 2022-08-24Version 1

Let $T$ be the map defined on $\N$ 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 $(\N, 2^{\N}, \nu, T)$ where $\nu$ is a finite measure equivalent to the counting measure. Define the operator $U : L^1(\nu) \rightarrow L^1(\nu) $ by $Uf = f\circ T. $ We prove that for each $n\in \N$ the set $\{T^k(n) : k\in\N\}$ is bounded. We 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 $L^1(\nu).$