Weak containment and Rokhlin entropy

TL;DR

This paper introduces a new concept of weak containment for joinings that leads to bounds on Rokhlin entropy and insights into Pinsker algebras and subgroup actions.

Contribution

It defines weak containment for joinings and demonstrates its implications for Rokhlin entropy inequalities and algebraic behavior under product and subgroup actions.

Findings

New bounds on Rokhlin entropy established.

Behavior of Pinsker algebras under direct products analyzed.

Rokhlin entropy of subgroup-restricted actions studied.

Abstract

We define a new notion of weak containment for joinings, and we show that this notion implies an inequality between relative Rokhlin entropies. This leads to new upper bounds to Rokhlin entropy. We also use this notion to study how Pinsker algebras behave under direct products, and we study the Rokhlin entropy of restricted actions of finite-index subgroups.