Collatz map as a non-singular transformation

TL;DR

This paper studies the Collatz map as a non-singular dynamical system, proving the existence of an invariant measure and convergence of averages, and relates the Collatz conjecture to measure-theoretic properties of the system.

Contribution

It establishes the existence of an invariant finite measure for the Collatz map and links the conjecture to the boundedness of an associated operator in measure theory.

Findings

Existence of an invariant finite measure for the Collatz map.

Convergence of time averages for functions in L^1.

Equivalence of the Collatz conjecture to a measure-theoretic operator condition.

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 $\{1,2\}.$