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Generic points in a characteristic class for amenable group actions are closed in the Besicovitch pseudometric

Sejal Babel, Martha Łącka, Marcel Mroczek

TL;DR

The paper extends the stability result of Kanigowski–Kułaga-Przymus–Lemańczyk–de la Rue from $\mathbb{Z}$-actions to general countable amenable group actions by showing that the set of points generic for some measure within a characteristic class is closed under the Besicovitch pseudometric $D_{B,\mathcal{F}}$ and that the corresponding ergodic measures are closed in the $\bar{\rho}$ metric. It develops and exploits the $\bar{\rho}$ metric as a Følner-independent proxy for Besicovitch convergence, establishing its equivalence with $\overline{D}_{B,\mathcal{F}}$ on ergodic measures for tempered two-sided Følner sequences. The results unify pointwise genericity and measure-theoretic stability in amenable-group dynamics, and the applications demonstrate how Besicovitch-approximations yield zero-entropy and discrete-spectrum phenomena via explicit constructions and known joinings/factor techniques. This provides a robust framework to study spectral and structural properties under metric limits in a broad dynamical setting with $G$-actions.

Abstract

We consider an action of a countable amenable group on a compact metric space, focusing on the set of generic points with respect to a fixed Følner sequence. For a given characteristic class, we prove that the set of points that are generic (along the Følner sequence) for some measure in this class is closed with respect to the Besicovitch pseudometric.

Generic points in a characteristic class for amenable group actions are closed in the Besicovitch pseudometric

TL;DR

The paper extends the stability result of Kanigowski–Kułaga-Przymus–Lemańczyk–de la Rue from -actions to general countable amenable group actions by showing that the set of points generic for some measure within a characteristic class is closed under the Besicovitch pseudometric and that the corresponding ergodic measures are closed in the metric. It develops and exploits the metric as a Følner-independent proxy for Besicovitch convergence, establishing its equivalence with on ergodic measures for tempered two-sided Følner sequences. The results unify pointwise genericity and measure-theoretic stability in amenable-group dynamics, and the applications demonstrate how Besicovitch-approximations yield zero-entropy and discrete-spectrum phenomena via explicit constructions and known joinings/factor techniques. This provides a robust framework to study spectral and structural properties under metric limits in a broad dynamical setting with -actions.

Abstract

We consider an action of a countable amenable group on a compact metric space, focusing on the set of generic points with respect to a fixed Følner sequence. For a given characteristic class, we prove that the set of points that are generic (along the Følner sequence) for some measure in this class is closed with respect to the Besicovitch pseudometric.

Paper Structure

This paper contains 12 sections, 17 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notation
  4. Amenable Groups
  5. Generic Points
  6. Factors and Joinings
  7. Characteristic classes
  8. The Upper Asymptotic Density
  9. The Besicovitch Pseudometric
  10. The $\bar{\rho}$ Metric on the Space of $G$-invariant Measures
  11. Main results
  12. Applications

Key Result

Theorem 2.1

Let $G$ be a countable discrete group, and let $(X,G,\mu)$ and $(Y,G,\nu)$ be two measure-preserving systems. Suppose there exists a joining $\lambda \in J(\mu, \nu)$ such that Then $(Y,G,\nu)$ is a factor of $(X,G,\mu)$.

Figures (1)

  • Figure 1: Central region of the lattice with points visible from the origin marked as larger blue dots.

Theorems & Definitions (35)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 25 more