Coboundaries of commuting Borel automorphisms

TL;DR

This paper investigates the properties of coboundaries in commuting Borel automorphisms, demonstrating differences in their coboundary sets and establishing a variant of Rokhlin's Lemma for Borel -actions.

Contribution

It shows that commuting automorphisms generating a free -action have distinct sets of bounded coboundaries and proves a weaker Rokhlin Lemma for Borel -actions.

Findings

Commuting automorphisms with free -actions have different bounded coboundary sets.

A weaker form of Rokhlin Lemma is established for Borel -actions.

Abstract

We show that if $S,T$ are two commuting automorphisms of standard Borel space such that they generate a free Borel $\Z^2$-action then $S$ and $T$ do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel $\Z^d$-actions.