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Metric entropy and homoclinic growth rate

Gang Liao, Jing Wei

TL;DR

The paper quantifies chaos via a quantitative link between metric entropy and homoclinic growth, proving a log-type lower bound for hyperbolic measures and, when maximal entropy is present, an exp-type growth bound for transverse homoclinic points. It develops a unified framework combining Pesin theory, horseshoes, and symbolic dynamics to translate entropy into homoclinic complexity, and demonstrates that Newhouse-domain dynamics can exhibit super-exponential growth of homoclinic points under perturbations. These results extend Mendoza’s two-dimensional insights to higher dimensions and provide a foundation for Katok-type conjectures in non-uniformly hyperbolic settings.

Abstract

In this paper, we investigate the relationship between chaos and homoclinic orbits from a quantitative perspective. Let f be a C^r diffeomorphism (r > 1) on a compact Riemannian manifold preserving an ergodic hyperbolic measure. We show that the homoclinic growth rate is bounded below by the metric entropy. This result generalizes the work of Mendoza from surfaces to higher-dimensional systems from a measure-theoretic viewpoint. We also examine the sharpness of this estimate by demonstrating that, in the Newhouse domain, C^r-generic diffeomorphisms exhibit a superexponential growth in the number of homoclinic points.

Metric entropy and homoclinic growth rate

TL;DR

The paper quantifies chaos via a quantitative link between metric entropy and homoclinic growth, proving a log-type lower bound for hyperbolic measures and, when maximal entropy is present, an exp-type growth bound for transverse homoclinic points. It develops a unified framework combining Pesin theory, horseshoes, and symbolic dynamics to translate entropy into homoclinic complexity, and demonstrates that Newhouse-domain dynamics can exhibit super-exponential growth of homoclinic points under perturbations. These results extend Mendoza’s two-dimensional insights to higher dimensions and provide a foundation for Katok-type conjectures in non-uniformly hyperbolic settings.

Abstract

In this paper, we investigate the relationship between chaos and homoclinic orbits from a quantitative perspective. Let f be a C^r diffeomorphism (r > 1) on a compact Riemannian manifold preserving an ergodic hyperbolic measure. We show that the homoclinic growth rate is bounded below by the metric entropy. This result generalizes the work of Mendoza from surfaces to higher-dimensional systems from a measure-theoretic viewpoint. We also examine the sharpness of this estimate by demonstrating that, in the Newhouse domain, C^r-generic diffeomorphisms exhibit a superexponential growth in the number of homoclinic points.

Paper Structure

This paper contains 15 sections, 2 theorems, 98 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homoclinic relation and Homoclinic point of order-$n$
  4. log-type growth
  5. Pesin set
  6. Admissible manifolds
  7. Closing lemma
  8. $\lambda$-lemma
  9. Katok's entropy formula
  10. Entropy and horseshoes
  11. Proof of Theorem \ref{['theorem a']}
  12. exp-type growth
  13. Symbolic dynamics and homoclinic point
  14. Proof of Theorem \ref{['theorem b']}
  15. Super-exponential growth

Key Result

Corollary 1

Let $f$ be a $C^{\infty}$ diffeomorphism on a compact surface $M$ with $h_{\operatorname{top}}(f)>0$. Then there exists a hyperbolic point $p$ and small $\varepsilon>0$,

Figures (5)

  • Figure 1: The transversely homoclinic point $t$ of periodic point $p$.
  • Figure 2: Homoclinic point $f^{-(n-r)}t$ of $\varepsilon$ order-$n$ of the periodic point $p$
  • Figure 3: The construction of the new homoclinic points.
  • Figure 4: Construct an interval of tangencies between $W^u(p_{\hat{g}})$ and $W^s(p_{\hat{g}})$
  • Figure 5: By a small $C^r$-perturbation, the inequality (\ref{['N_1']}) still holds.

Theorems & Definitions (12)

  • Conjecture 1
  • Corollary 1
  • Remark 1
  • Conjecture 2
  • proof
  • proof : Proof of claim
  • Proposition 1
  • proof : Proof of Proposition \ref{['prop 3']}
  • proof
  • proof : Proof of Corollary 1
  • ...and 2 more