Metric entropy and homoclinic growth rate

TL;DR

This paper explores the connection between chaos and homoclinic orbits in dynamical systems, establishing a lower bound for homoclinic growth rate based on metric entropy and analyzing generic behaviors in higher dimensions.

Contribution

It generalizes Mendoza's results to higher-dimensional systems and measure-theoretic contexts, providing new bounds and insights into homoclinic growth rates.

Findings

Homoclinic growth rate is bounded below by metric entropy.

In the Newhouse domain, generic diffeomorphisms show superexponential homoclinic point growth.

The results extend from surfaces to higher-dimensional systems.

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.