Infinite Partitions and Rokhlin Towers

TL;DR

This paper constructs special partitions and extensions for non-periodic measure-preserving transformations, analyzes Rokhlin towers, and explores properties of uniform martingales with applications to entropy and isomorphism.

Contribution

It introduces a method to generate partitions that encode transformations efficiently and extends transformations to uniform martingales on infinite alphabets, with new insights into Rokhlin towers.

Findings

Constructed partitions that generate transformations with local information

Extended transformations to uniform martingales on infinite alphabets

Provided a detailed analysis of Rokhlin towers and their properties

Abstract

We find a countable partition $P$ on\textbf{} a Lebesgue space, labeled $\{1,2,3...$\}, for any non-periodic measure preserving transformation $T$ such that $P$ generates $T$ and for the $T,P$ process, if you see an $n$ on time -1 then you only have to look at times $-n,1-n,...-1$ to know the positive integer $i$ to put at time 0. We alter that proof to extend every non-periodic $T$ to a uniform martingale (i.e. continuous $g$ function) on an infinite alphabet. If $T$ has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an i.i.d. process on $\mathbb{Z}^2$ when you read from right to left but where each column determines the next if you read left to right.