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A Hofer-like Metric on the Space of Anosov Flows

Stéphane Tchuiaga

TL;DR

This work introduces Hofer-like metrics $d_{\mathrm{An},V}$ on the space of Anosov vector fields, built from deformation paths measured in $C^k$ or Sobolev norms to yield a dynamically meaningful geometry. It proves completeness (in $C^r$ and $H^k$ settings), diffeomorphism naturality, and robust links between metric distance and variations of entropy, Lyapunov exponents, SRB measures, and thermodynamic quantities, including differentiability under strong regularity. The paper also analyzes local geodesics in the $H^k$ setting, Lipschitz stability for pressure and spectral gaps, and the moduli-space structure via a proposed slice framework; it further introduces Topological Anosov Flows as non-smooth limits captured by $d_{\mathrm{An}}$-Cauchy generator sequences. These results establish a quantitative, geometric approach to stability, classification, and potential extensions of Anosov dynamics, with broad implications for hyperbolic theory and thermodynamic formalism. Conjectures and future directions point toward global geometry, curvature interpretations, and a broader understanding of non-smooth hyperbolic phenomena through these metrics.

Abstract

This paper develops a family of Hofer-like metrics ($\dAnV{V}$) on the space of Anosov vector fields $\An(M)$, providing dynamically relevant distances based on the cost of deformation paths using $\Ck{k}$ or Sobolev $\SobolevHk{k}$ norms. We establish fundamental properties, including completeness for $V= C^r (r \ge 1)$ or $H^k (k > \dim(M)/2+1)$, and naturality under diffeomorphisms. We show the utility of these metrics by proving quantitative stability results: proximity in $\dAnV{V}$ implies controlled variation of essential dynamical invariants, including topological entropy, Lyapunov exponents, SRB measures, thermodynamic pressure, spectral gaps (mixing rates), and zeta functions. Sufficient regularity ensures local Lipschitz continuity and Fréchet differentiability, connecting the metric structure to linear response formulas, particularly for pressure, exponents, and the spectral gap. While Sobolev metrics yield locally flat geometry with straight line geodesics, the framework is broadly applicable. We explore implications for the moduli space of Anosov flows, including stability of invariants and the framework for local slice theorems. Furthermore, we introduce \emph{Topological Anosov Flows}, defined via simultaneous uniform flow convergence and $\dAn$ metric convergence of the generating fields. This new class aims to capture essential hyperbolic features in non-smooth settings. Overall, the proposed metrics offer several geometric perspectives for analyzing the stability, classification, and possible extensions of Anosov dynamics.

A Hofer-like Metric on the Space of Anosov Flows

TL;DR

This work introduces Hofer-like metrics on the space of Anosov vector fields, built from deformation paths measured in or Sobolev norms to yield a dynamically meaningful geometry. It proves completeness (in and settings), diffeomorphism naturality, and robust links between metric distance and variations of entropy, Lyapunov exponents, SRB measures, and thermodynamic quantities, including differentiability under strong regularity. The paper also analyzes local geodesics in the setting, Lipschitz stability for pressure and spectral gaps, and the moduli-space structure via a proposed slice framework; it further introduces Topological Anosov Flows as non-smooth limits captured by -Cauchy generator sequences. These results establish a quantitative, geometric approach to stability, classification, and potential extensions of Anosov dynamics, with broad implications for hyperbolic theory and thermodynamic formalism. Conjectures and future directions point toward global geometry, curvature interpretations, and a broader understanding of non-smooth hyperbolic phenomena through these metrics.

Abstract

This paper develops a family of Hofer-like metrics () on the space of Anosov vector fields , providing dynamically relevant distances based on the cost of deformation paths using or Sobolev norms. We establish fundamental properties, including completeness for or , and naturality under diffeomorphisms. We show the utility of these metrics by proving quantitative stability results: proximity in implies controlled variation of essential dynamical invariants, including topological entropy, Lyapunov exponents, SRB measures, thermodynamic pressure, spectral gaps (mixing rates), and zeta functions. Sufficient regularity ensures local Lipschitz continuity and Fréchet differentiability, connecting the metric structure to linear response formulas, particularly for pressure, exponents, and the spectral gap. While Sobolev metrics yield locally flat geometry with straight line geodesics, the framework is broadly applicable. We explore implications for the moduli space of Anosov flows, including stability of invariants and the framework for local slice theorems. Furthermore, we introduce \emph{Topological Anosov Flows}, defined via simultaneous uniform flow convergence and metric convergence of the generating fields. This new class aims to capture essential hyperbolic features in non-smooth settings. Overall, the proposed metrics offer several geometric perspectives for analyzing the stability, classification, and possible extensions of Anosov dynamics.

Paper Structure

This paper contains 28 sections, 20 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Analogy with the Hofer Metric
  4. Construction of the Metric on $\mathrm{An}(M)$
  5. Properties
  6. Invariance Properties of the Hofer-like Metric
  7. Completeness
  8. Anosov Flows Approximations
  9. Discussion of Metric Variants
  10. Dynamical Significance and Stability Results
  11. Continuity of Topological Entropy
  12. Existence and Properties of Minimizing Geodesics
  13. Stability of Lyapunov Exponents
  14. Continuity of SRB Measures and Thermodynamic Quantities
  15. Convergence of Periodic Orbit Data and Zeta Functions
  16. ...and 13 more sections

Key Result

Proposition 5.2

Let $M$ be a compact smooth manifold. For any auxiliary Riemannian metric $g$ on $M$, let $f: M \to M$ be a smooth ($C^k$, $k \ge r+1$) diffeomorphism, and let $f^*g$ denote the pullback metric. Let $V$ represent a type of norm on $C^r(TM)$ (such as $C^l$ for $0 \le l \le r$, or Sobolev $H^l$ for $l satisfies the isometry property relating the norms based on $g$ and $f^*g$: Let $d_{\mathrm{An}, V

Theorems & Definitions (78)

  • Definition 2.1: Anosov Flow
  • Definition 4.1: Paths in $\mathrm{An}(M)$
  • Definition 4.2: Velocity and $\mathcal{C}^{0}$ Norm
  • Definition 4.3: Length of a Path
  • Definition 4.4: Hofer-like Metric $d_{\mathrm{An}}$
  • Remark 5.1: Non-Degeneracy in Specific Settings
  • Proposition 5.2: Diffeomorphism Pushforward Induces Isometry Between Metric Spaces
  • proof
  • Remark 5.3
  • Corollary 5.4: Invariance under Isometries
  • ...and 68 more