Independence and Alpern Multitowers

TL;DR

This paper proves that for any invertible, ergodic, aperiodic measure-preserving transformation and any finite partition, it is always possible to construct an Alpern multitower with a base independent of that partition.

Contribution

It introduces a method to construct Alpern multitowers with bases independent of any given finite partition in ergodic theory.

Findings

Existence of independent Alpern multitowers for any finite partition

Construction applicable to any invertible, ergodic, aperiodic transformation

Enhances understanding of structure in measure-preserving systems

Abstract

Let $T$ be any invertible, ergodic, aperiodic measure-preserving transformation of a Lebesgue probability space $(X, \calB, \mu)$, and \P\, any finite measurable partition of $X$. We show that a (finite) Alpern multitower may always be constructed whose base is independent of \P.