Entropy of group actions beyond uniform lattices

TL;DR

This paper extends entropy concepts from discrete to topological groups, revealing key differences and establishing invariants, pressure, and theorems for amenable unimodular groups, with implications for group actions beyond lattices.

Contribution

It generalizes entropy and pressure notions to amenable topological groups, contrasting discrete and non-discrete cases, and connects these to uniform lattice approaches.

Findings

The first entropy concept collapses to 0 in non-discrete groups.

The second entropy yields a well-behaved invariant for amenable unimodular groups.

A Goodwyn-type theorem and equivalence with lattice-based approaches are established.

Abstract

Entropy of measure preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are the ones given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein-Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein-Weiss lemma due to Gromov.