Infinite Partitions and Rokhlin Towers

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.