Hyperbolic periodic points for chain hyperbolic homoclinic classes

TL;DR

This paper proves that in certain chain hyperbolic homoclinic classes with one-dimensional centers, the number of hyperbolic periodic points grows at a rate equal to the topological entropy, and hyperbolic measures are dense.

Contribution

It establishes closing properties for chain hyperbolic homoclinic classes with one-dimensional centers and links periodic point growth to topological entropy.

Findings

Growth rate of hyperbolic periodic points equals topological entropy

Hyperbolic periodic measures are dense among invariant measures

Closing properties are established for the classes studied

Abstract

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.