Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Independence and mean sensitivity in minimal systems under group actions

Chunlin Liu, Leiye Xu, Shuhao Zhang

TL;DR

The paper addresses how regularity, independence, and mean sensitivity interact in minimal systems under group actions, introducing an interior-saturation framework for hyperspace dynamics. It proves that under either incontractibility or local Bronstein conditions with an invariant measure, fibers of the maximal equicontinuous factor carry IT-sets for a full-measure set of base points, with strong confirmations in virtually nilpotent and proximal-extension cases. In amenable-group settings under local Bronstein, it establishes parallel mean-sensitivity results and shows that mean-sensitive tuples are IT-tuples, with stronger statements when $\pi_{eq}$ is open, unifying several sensitivity notions (IT, IN, sequence entropy, and weak mean-sensitivity). The results yield optimal bounds on ergodic measures in certain extensions and advance the understanding of entropy and independence in non-abelian and distal settings, providing effective tools for non-abelian dynamics and broad applicability in mean-based sensitivity theory.

Abstract

In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independence. In particular, the following two types of minimal systems are applicable to our result: (1) The acting group of the minimal system is a virtually nilpotent group. (2) The minimal system is a proximal extension of its maximal equicontinuous factor and admits an invariant Borel probability measure. Items (1) and (2) correspond to Conjectures 1 and 2 from Huang, Lian, Shao, and Ye (J. Funct. Anal., 2021); item (1) verifies Conjecture 1 in the virtually nilpotent case, and item (2) gives an affirmative answer to Conjecture 2. In the second part, for a minimal system acting by an amenable group, under the local Bronstein condition, we establish parallel results regarding weak mean sensitivity and establish that every mean-sensitive tuple is an IT-tuple.

Independence and mean sensitivity in minimal systems under group actions

TL;DR

The paper addresses how regularity, independence, and mean sensitivity interact in minimal systems under group actions, introducing an interior-saturation framework for hyperspace dynamics. It proves that under either incontractibility or local Bronstein conditions with an invariant measure, fibers of the maximal equicontinuous factor carry IT-sets for a full-measure set of base points, with strong confirmations in virtually nilpotent and proximal-extension cases. In amenable-group settings under local Bronstein, it establishes parallel mean-sensitivity results and shows that mean-sensitive tuples are IT-tuples, with stronger statements when is open, unifying several sensitivity notions (IT, IN, sequence entropy, and weak mean-sensitivity). The results yield optimal bounds on ergodic measures in certain extensions and advance the understanding of entropy and independence in non-abelian and distal settings, providing effective tools for non-abelian dynamics and broad applicability in mean-based sensitivity theory.

Abstract

In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independence. In particular, the following two types of minimal systems are applicable to our result: (1) The acting group of the minimal system is a virtually nilpotent group. (2) The minimal system is a proximal extension of its maximal equicontinuous factor and admits an invariant Borel probability measure. Items (1) and (2) correspond to Conjectures 1 and 2 from Huang, Lian, Shao, and Ye (J. Funct. Anal., 2021); item (1) verifies Conjecture 1 in the virtually nilpotent case, and item (2) gives an affirmative answer to Conjecture 2. In the second part, for a minimal system acting by an amenable group, under the local Bronstein condition, we establish parallel results regarding weak mean sensitivity and establish that every mean-sensitive tuple is an IT-tuple.
Paper Structure (18 sections, 28 theorems, 124 equations)

This paper contains 18 sections, 28 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Preliminaries
  5. Subset of group
  6. Topological dynamical system
  7. System on the Hyperspace
  8. Ellis semigroup
  9. Interior saturation property
  10. Notation
  11. Sets satisfying interior saturation property
  12. Proof of Proposition \ref{['prop-2']}
  13. IT-set
  14. Statement of main result and corollaries
  15. Proof of Theorem \ref{['main1']}
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $(X,G)$ be a minimal tds, and let $\pi_{eq}: X\to X_{eq}$ be the factor map to its maximal equicontinuous factor. Assume one of the following conditions holds: Then, there exists a dense $G_\delta$ subset $Z\subset X_{eq}$ with $\nu_{eq}(Z)=1$ such that $\pi_{eq}^{-1}(y)$ is an IT-set for every $y\in Z$.

Theorems & Definitions (60)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 50 more