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Ergodic measures with infinite entropy

Eleonora Catsigeras, Serge Troubetzkoy

TL;DR

The paper proves that generic continuous maps on compact manifolds (and generic homeomorphisms when $\dim M\ge2$) admit ergodic invariant measures with metric entropy $h_{\mu}(f)=+\infty$ and that these measures can be mixing for an iterate $f^p$. It achieves this by constructing a family of model maps ${\mathcal H}$ on $D^m$ with a Cantor invariant set supporting an invariant measure of infinite entropy (Lemma Main) and showing that return maps to periodic shrinking boxes realize these models, yielding sequences of infinite-entropy ergodic measures that can converge to a zero-entropy limit. The work also demonstrates strong non upper semi-continuity of entropy in the $C^0$-generic setting and provides a framework to realize local complex dynamics at arbitrarily small scales that influence global behavior. Additionally, it outlines good sequences of periodic shrinking boxes enabling weak$^*$-convergence of high-entropy measures to zero-entropy limits and poses several natural questions about regularity and extensions to non-manifold spaces.

Abstract

We construct ergodic probability measures with infinite metric entropy for typical continuous maps and homeomorphisms on compact manifolds. We also construct sequences of such measures that converge to a zero-entropy measure.

Ergodic measures with infinite entropy

TL;DR

The paper proves that generic continuous maps on compact manifolds (and generic homeomorphisms when ) admit ergodic invariant measures with metric entropy and that these measures can be mixing for an iterate . It achieves this by constructing a family of model maps on with a Cantor invariant set supporting an invariant measure of infinite entropy (Lemma Main) and showing that return maps to periodic shrinking boxes realize these models, yielding sequences of infinite-entropy ergodic measures that can converge to a zero-entropy limit. The work also demonstrates strong non upper semi-continuity of entropy in the -generic setting and provides a framework to realize local complex dynamics at arbitrarily small scales that influence global behavior. Additionally, it outlines good sequences of periodic shrinking boxes enabling weak-convergence of high-entropy measures to zero-entropy limits and poses several natural questions about regularity and extensions to non-manifold spaces.

Abstract

We construct ergodic probability measures with infinite metric entropy for typical continuous maps and homeomorphisms on compact manifolds. We also construct sequences of such measures that converge to a zero-entropy measure.

Paper Structure

This paper contains 6 sections, 31 theorems, 193 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Construction of the family of models.
  3. Infinite metric entropy and mixing property of the models.
  4. Periodic Shrinking Boxes
  5. Good sequences of periodic shrinking boxes
  6. Open questions

Key Result

Theorem 1

The generic map $f \in C^0(M)$ has an ergodic Borel probability measure $\mu$ such that $h_{\mu}(f) = + \infty$ and there exists $p \ge 1$ such that $\mu$ is mixing for the map $f^p$.

Figures (5)

  • Figure 1: The atom $A$ of generation 0 and two atoms $B, C$ of generation 1 for a map $\Phi$ of $D^2$.
  • Figure 2: An atom $A$ of generation 0, two atoms $B,C$ of generation 1, and $16$ atoms of generation 2. In particular the two atoms $G, H$ of generation 2 satisfy $\Gamma_2(C,B,C) = \{G, H\}$.
  • Figure 3: Construction of $g$ near $f$ with a periodic shrinking box $K$ for $g$.
  • Figure 4: Perturbation $g$ of $f$ such that $g^p|_K = \Phi$.
  • Figure 5: Construction of a good sequence of periodic shrinking boxes.

Theorems & Definitions (80)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 70 more