Hyperbolic Periodic Points and Hyperbolic Measures with Dominated Splitting

TL;DR

This paper proves exponential shadowing and closing lemmas for hyperbolic measures with dominated splitting, establishing the density of hyperbolic periodic points and deriving classical results like Livshitz Theorem.

Contribution

It introduces exponential shadowing and closing lemmas for hyperbolic measures with dominated splitting, extending classical hyperbolic theory results.

Findings

Existence of hyperbolic periodic points with positive measure unstable manifolds.

Density of hyperbolic periodic points in the support of the measure.

Derivation of Livshitz Theorem for the setting.

Abstract

In this paper we consider a non-atomic invariant hyperbolic measure $\mu$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $\mu$ a.e. points. We show an \textit{exponentially} shadowing and an \textit{exponentially} closing lemma, and as applications we show two classical results. One is that there exists a hyperbolic periodic point such that the closure of its unstable manifold has \textit{positive} measure and it has a homoclinic point from which one can deduce a horseshoe. Moreover, such hyperbolic periodic points are dense in the support $supp(\mu)$ of the given hyperbolic measure. Another is to show Livshitz Theorem.