An Alpern tower independent of a given partition

TL;DR

This paper presents a method to construct Alpern towers of any height with bases independent of a given finite measurable partition in measure-preserving dynamical systems.

Contribution

It introduces a construction technique for Alpern towers with bases independent of specified partitions, extending the tools available in ergodic theory.

Findings

Constructed Alpern towers of arbitrary height with independent bases.

Provided a method ensuring the tower's base is independent of a given partition.

Demonstrated the existence of such towers in measure-preserving transformations.

Abstract

Given a measure-preserving transformation $T$ of a probability space $(X, \mathcal B, \mu)$ and a finite measurable partition $P$ of $X$, we show how to construct an Alpern tower of any height whose base is independent of the partition $P$. That is, given $N \in \N$, there exists a Rohlin tower of height $N$, with base $B$ and error set $E$, so that $B$ is independent of $P$, and $T(E) \subset B$.