Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures

TL;DR

This paper constructs horseshoes for certain smooth maps with hyperbolic measures, linking periodic point growth rates to hyperbolic entropy, thus advancing understanding of dynamical complexity in smooth systems.

Contribution

It introduces a method to construct horseshoes for $ ext{C}^{1+eta}$ maps with hyperbolic measures, connecting periodic point growth to hyperbolic entropy.

Findings

Exponential growth rate of periodic points ≥ measure-theoretic entropy

Growth rate of hyperbolic periodic points equals hyperbolic entropy

Horseshoes constructed for $ ext{C}^{1+eta}$ maps with hyperbolic measures

Abstract

We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.