Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropy of group actions beyond uniform lattices

Till Hauser, Friedrich Martin Schneider

TL;DR

This work develops an intrinsic entropy theory for actions of amenable topological groups, contrasting Ollagnier entropy (via thin Følner nets) with Ornstein–Weiss entropy (via van Hove nets and Delone sets) and extending both to non-discrete settings. It proves a topological Ollagnier lemma guaranteeing convergence of normalized entropies and derives that Ollagnier entropy vanishes for non-discrete metrizable or locally compact groups, while Ornstein–Weiss entropy remains a robust invariant, independent of averaging nets and Delone choices. The paper then defines Ornstein–Weiss topological pressure, establishes a Goodwyn-type inequality, and shows how pressure and entropy behave under restriction to uniform lattices, thereby linking the continuous-group theory to classical discrete cases when a uniform lattice exists. Finally, it constructs topological models for measure-preserving actions and develops the algebraic-topological framework needed to compare entropy notions across representations, with potential applications in aperiodic order and ergodic theory. Overall, the results provide a comprehensive, intrinsic approach to entropy and pressure for amenable topological group actions and clarify when each notion yields meaningful invariants.

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.

Entropy of group actions beyond uniform lattices

TL;DR

This work develops an intrinsic entropy theory for actions of amenable topological groups, contrasting Ollagnier entropy (via thin Følner nets) with Ornstein–Weiss entropy (via van Hove nets and Delone sets) and extending both to non-discrete settings. It proves a topological Ollagnier lemma guaranteeing convergence of normalized entropies and derives that Ollagnier entropy vanishes for non-discrete metrizable or locally compact groups, while Ornstein–Weiss entropy remains a robust invariant, independent of averaging nets and Delone choices. The paper then defines Ornstein–Weiss topological pressure, establishes a Goodwyn-type inequality, and shows how pressure and entropy behave under restriction to uniform lattices, thereby linking the continuous-group theory to classical discrete cases when a uniform lattice exists. Finally, it constructs topological models for measure-preserving actions and develops the algebraic-topological framework needed to compare entropy notions across representations, with potential applications in aperiodic order and ergodic theory. Overall, the results provide a comprehensive, intrinsic approach to entropy and pressure for amenable topological group actions and clarify when each notion yields meaningful invariants.

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.
Paper Structure (32 sections, 45 theorems, 182 equations)

This paper contains 32 sections, 45 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Compact Hausdorff spaces
  5. Topological groups
  6. Actions of topological groups
  7. Amenability
  8. Geometric notions
  9. van Hove nets
  10. Uniform lattices and van Hove nets
  11. Combinatorial properties of set functions
  12. The Ollagnier lemma
  13. Continuity of set functions
  14. Proof of Ollagnier's lemma for topological groups
  15. The Ornstein-Weiss lemma
  16. ...and 17 more sections

Key Result

Theorem 1

Let $G$ be an amenable topological group, and let $f\colon \mathcal{F}(G)\to \mathbb{R}_{\geq 0}$ be a continuous, right-invariant function satisfying Shearer's inequality. Then, for any thin Fø lner net $(F_i)_{i\in I}$ in $G$, the following limit exists and satisfies

Theorems & Definitions (91)

  • Theorem : Topological version of Ollagnier's lemma; Theorem \ref{['the:ollagnierslemma']}
  • Theorem
  • Theorem
  • Theorem : Ornstein-Weiss lemma; Theorem \ref{['the:Ornstein-Weisslemma']}
  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3: schneider2016folner, Theorem 4.5
  • ...and 81 more