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On the Expansiveness of Invariant Measures under Pseudogroups

A. Arbieto, L. Segantim, J. Siqueira

Abstract

In this paper, we define and study weak expansive and expansive measures for pseudogroups, these two notions appear when analyzing the role of the generating set. We investigate the relations between such properties. We also provide a criterion for a measure to be weak expansive through the positivity of its entropy, generalizing the work of Arbieto and Morales. We also show that in some settings equicontinuous pseudogroups have no expansive measures.

On the Expansiveness of Invariant Measures under Pseudogroups

Abstract

In this paper, we define and study weak expansive and expansive measures for pseudogroups, these two notions appear when analyzing the role of the generating set. We investigate the relations between such properties. We also provide a criterion for a measure to be weak expansive through the positivity of its entropy, generalizing the work of Arbieto and Morales. We also show that in some settings equicontinuous pseudogroups have no expansive measures.
Paper Structure (12 sections, 28 theorems, 70 equations)

This paper contains 12 sections, 28 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudogroups
  4. Generating Sets
  5. Dynamical Balls
  6. Good Generating Sets
  7. Ergodic Theory of Pseudogroups
  8. Expansive Measures
  9. Conjugacy Properties
  10. Homogeneous Measures and Expansiveness
  11. Equicontinuity
  12. Acknowledgements

Key Result

Theorem 1

Let $\mathcal{G}$ a good pseudogroup, $\mathcal{G}_{1}$ a good generating set and $\mathcal{G}_{2}$ the compacted generating set. Then, there exists $\rho> 0$ such that if a measure $\mu$ is $(\mathcal{G}, \mathcal{G}_{2})$-weakly expansive with constant $\rho > 0$ then $\mu$ is $(\mathcal{G}, \math

Theorems & Definitions (76)

  • Theorem 1
  • Theorem 2: Criterion for Weakly Expansive Measures
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Example 5
  • Example 6
  • Lemma 7
  • ...and 66 more