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The maximal mean equicontinuous factor via regional mean sensitivity

Till Hauser, Chunlin Liu

Abstract

For actions of amenable groups, mean equicontinuity-a natural relaxation of equicontinuity obtained by averaging metrics along orbits-is well known to yield a maximal mean equicontinuous factor. In 2021, Li and Yu introduced the notion of weak sensitivity in the mean for actions of $\mathbb{Z}$ to gain a deeper understanding of this phenomenon, building on earlier work by Qiu and Zhao. We demonstrate that this relation is insufficient for actions of non-Abelian groups. To overcome this limitation, we introduce the regional mean sensitive relation, which more precisely captures the dynamical behaviour underlying the maximal mean equicontinuous factor. We discuss its fundamental properties and highlight its advantages in the non-Abelian setting. In particular, we show that mean equicontinuity is equivalent to the nonexistence of non-diagonal regional mean sensitive pairs. For this, we work in the context of actions of $σ$-compact and locally compact amenable groups.

The maximal mean equicontinuous factor via regional mean sensitivity

Abstract

For actions of amenable groups, mean equicontinuity-a natural relaxation of equicontinuity obtained by averaging metrics along orbits-is well known to yield a maximal mean equicontinuous factor. In 2021, Li and Yu introduced the notion of weak sensitivity in the mean for actions of to gain a deeper understanding of this phenomenon, building on earlier work by Qiu and Zhao. We demonstrate that this relation is insufficient for actions of non-Abelian groups. To overcome this limitation, we introduce the regional mean sensitive relation, which more precisely captures the dynamical behaviour underlying the maximal mean equicontinuous factor. We discuss its fundamental properties and highlight its advantages in the non-Abelian setting. In particular, we show that mean equicontinuity is equivalent to the nonexistence of non-diagonal regional mean sensitive pairs. For this, we work in the context of actions of -compact and locally compact amenable groups.
Paper Structure (23 sections, 37 theorems, 82 equations)

This paper contains 23 sections, 37 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Probability measures
  4. Følner sequences and amenability
  5. Upper densities
  6. Actions
  7. Cesàro averages
  8. Invariant Borel probability measures and generic points
  9. The maximal support
  10. Mean equicontinuity
  11. The interplay between the relations
  12. The interplay of $\mathrm{S}_\mathrm{rjms}^{\mathcal{F}}$ and $\mathrm{S}_\mathrm{wsm}^{\mathcal{F}}$
  13. Basic properties of $\mathrm{Q}_\mathrm{rms}^{(\mathcal{F})}$ and $\mathrm{S}_\mathrm{rjms}^{(\mathcal{F})}$
  14. The interplay of $\mathrm{Q}_\mathrm{rms}^{(\mathcal{F})}$ and $\mathrm{S}_\mathrm{rjms}^{(\mathcal{F})}$
  15. Characterisation via invariant probability measures on $X^2$
  16. ...and 8 more sections

Key Result

Theorem 1

Let $(X,G)$ be an action of a locally compact and $\sigma$-compact amenable group on a compact metric space and $\mathcal{F}$ be a Følner sequence in $G$. We have $\mathrm{R}_\mathrm{me}^\mathcal{F}(X)=\langle \mathrm{Q}_\mathrm{rms}^\mathcal{F}(X) \rangle$.

Theorems & Definitions (93)

  • Definition 1.1
  • Example
  • Definition 1.2: $\mathcal{F}$-regionally mean sensitive pairs
  • Theorem
  • Definition 1.3
  • Theorem
  • Corollary
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 83 more