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Independence, sequence entropy and mean sensitivity for invariant measures

Chunlin Liu, Leiye Xu, Shuhao Zhang

TL;DR

The paper investigates how independence, sequence entropy, and mean sensitivity relate for measure-preserving actions of countable infinite groups on compact spaces. It develops a localization framework via disintegration over the Kronecker factor to connect measure-theoretic sequence entropy with IT tuples, and proves that SE_K^μ(X,G) ⊂ IT_K(X,G). When the acting group is amenable and μ is ergodic, the work establishes a strong equivalence: along tempered Følner sequences, the μ-sequence entropy K-tuples, μ-mean sensitive K-tuples, and μ-sensitive-in-the-mean K-tuples all coincide. A complementary result provides an upper bound h_μ^*(G) ≤ log(K−1) in the absence of essential SE-tuples, with the converse holding in the ergodic amenable setting. Overall, the work extends local entropy theory to non-abelian amenable group actions, linking entropy and mean sensitivity through the Kronecker structure and providing a toolkit for pinpointing where complexity arises in the phase space.

Abstract

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered Følner sequence, and the sensitive in the mean tuples along some tempered Følner sequence coincide.

Independence, sequence entropy and mean sensitivity for invariant measures

TL;DR

The paper investigates how independence, sequence entropy, and mean sensitivity relate for measure-preserving actions of countable infinite groups on compact spaces. It develops a localization framework via disintegration over the Kronecker factor to connect measure-theoretic sequence entropy with IT tuples, and proves that SE_K^μ(X,G) ⊂ IT_K(X,G). When the acting group is amenable and μ is ergodic, the work establishes a strong equivalence: along tempered Følner sequences, the μ-sequence entropy K-tuples, μ-mean sensitive K-tuples, and μ-sensitive-in-the-mean K-tuples all coincide. A complementary result provides an upper bound h_μ^*(G) ≤ log(K−1) in the absence of essential SE-tuples, with the converse holding in the ergodic amenable setting. Overall, the work extends local entropy theory to non-abelian amenable group actions, linking entropy and mean sensitivity through the Kronecker structure and providing a toolkit for pinpointing where complexity arises in the phase space.

Abstract

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered Følner sequence, and the sensitive in the mean tuples along some tempered Følner sequence coincide.
Paper Structure (17 sections, 22 theorems, 98 equations)

This paper contains 17 sections, 22 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fø lner sequence
  4. Factor map
  5. Metric on the space of probability measures
  6. Disintegration of probability measures
  7. Kronecker factor
  8. Sequence entropy
  9. Mean sensitivity
  10. Proof of Theorem \ref{['main_abel']}
  11. Statement of Theorem \ref{['thm:IT_support']} and derivation of Theorem \ref{['main_abel']} from Theorem \ref{['thm:IT_support']}
  12. Proof of Theorem \ref{['thm:IT_support']}
  13. Proof of Theorem \ref{['thm-sequence-entropy']}
  14. Proof of Theorem \ref{['cor:se=sm']}
  15. Decomposition of ergodic measures
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(X, G)$ be a tds, and $\mu \in \mathcal{M}(X, G)$. Then where $SE_K^\mu(X,G)$ is the set of sequence entropy $K$-tuples for the measure $\mu$. Thus, combining this with the result in KH, one has where $SE_K(X,G)$ is the set of topological sequence entropy $K$-tuples.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 31 more