Hyperbolic Periodic Points and Hyperbolic Measures with Dominated Splitting

2010/11/28 by Xueting Tian, Tian, Xueting
Mathematics · Physics and Astronomy · #37C25 #37C50 #37D10 #37D25 #37D30 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1011.6011

openalex publication_date 2010/11/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper we consider a non-atomic invariant hyperbolic measure μ of a C¹ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on μ a.e. points. We show an exponentially shadowing and an 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 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(μ) of the given hyperbolic measure. Another is to show Livshitz Theorem.

Citations

• A generalized shadowing lemma

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