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Measure of maximal entropy for H-flows on non-compact manifolds

Anna Florio, Barbara Schapira, Anne Vaugon

TL;DR

The paper introduces H-flows as a robust framework for chaotic dynamics on non-compact manifolds, enabling a systematic comparison of entropy notions beyond compact settings. Under strong positive recurrence, quantified by an entropy-at-infinity being strictly smaller than the global entropy, the authors prove the existence of an invariant probability measure maximizing all standard entropy notions (KS, Katok, Brin-Katok, and Gurevic, with the variational entropy aligning). A central innovation is showing the Gurevic and chord entropies coincide, and that entropy at infinity can be controlled via periodic-orbit and chord counts, allowing a constructive approach using averages of periodic measures. The results extend the classical compact-manifold theory to a broad non-compact regime and lay groundwork for uniqueness/ergodicity questions, as well as for new families of non-compact examples. Overall, the work provides a unifying framework for entropy analysis on non-compact flows and establishes maximal-entropy measures under SPR, with clear implications for geodesic-type dynamics and beyond.

Abstract

In this work, we introduce a natural class of chaotic flows on non-compact manifolds, called H-flows, which includes geodesic flows on non-compact manifolds with pinched negative curvature. We show that, under the additional assumption, called strong positive recurrence, that their entropy at infinity is strictly smaller than the topological entropy, such flows admit an invariant probability measure maximizing entropy. In particular, we compare several notions of entropy in a non-compact setting.

Measure of maximal entropy for H-flows on non-compact manifolds

TL;DR

The paper introduces H-flows as a robust framework for chaotic dynamics on non-compact manifolds, enabling a systematic comparison of entropy notions beyond compact settings. Under strong positive recurrence, quantified by an entropy-at-infinity being strictly smaller than the global entropy, the authors prove the existence of an invariant probability measure maximizing all standard entropy notions (KS, Katok, Brin-Katok, and Gurevic, with the variational entropy aligning). A central innovation is showing the Gurevic and chord entropies coincide, and that entropy at infinity can be controlled via periodic-orbit and chord counts, allowing a constructive approach using averages of periodic measures. The results extend the classical compact-manifold theory to a broad non-compact regime and lay groundwork for uniqueness/ergodicity questions, as well as for new families of non-compact examples. Overall, the work provides a unifying framework for entropy analysis on non-compact flows and establishes maximal-entropy measures under SPR, with clear implications for geodesic-type dynamics and beyond.

Abstract

In this work, we introduce a natural class of chaotic flows on non-compact manifolds, called H-flows, which includes geodesic flows on non-compact manifolds with pinched negative curvature. We show that, under the additional assumption, called strong positive recurrence, that their entropy at infinity is strictly smaller than the topological entropy, such flows admit an invariant probability measure maximizing entropy. In particular, we compare several notions of entropy in a non-compact setting.

Paper Structure

This paper contains 48 sections, 55 theorems, 382 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Anosov flows on compact manifolds
  3. Dynamics on non-compact manifolds, motivations
  4. Our results
  5. TEXT-flows on non-compact manifolds
  6. Notations
  7. Bounds on the flow
  8. Topological transitivity
  9. Closing lemma
  10. Expansivity
  11. Shadowing properties
  12. TEXT-flows
  13. Properties of TEXT-flows
  14. Entropies
  15. Kolmogorov-Sinai entropy
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $\varphi\colon M\to M$ be a $H$-flow on a Riemannian manifold $(M,g)$ such that $h_\mathrm{Gur}^\infty(\varphi)<h_\mathrm{Gur}(\varphi)$. Then, there exists a $\varphi$-invariant probability measure $m_\mathrm{max}$ on $M$ maximizing entropies:

Figures (14)

  • Figure 2.1: Closing Lemma
  • Figure 2.2: Finite exact shadowing for $N=2$
  • Figure 2.3: Proof of Uniform Transitivity
  • Figure 2.4: Multiple closing Lemma for $N=2$
  • Figure 2.5: Uniform multiple closing Lemma for $N=2$
  • ...and 9 more figures

Theorems & Definitions (134)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2: Transitivity
  • Definition 2.3: Closing lemma
  • Definition 2.4: Expansivity
  • Proposition 2.5
  • proof
  • Definition 2.6: Finite exact shadowing
  • Definition 2.7: Local product structure
  • ...and 124 more