Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies

2003/06/02 by Yongluo Cao, Cao, Yongluo, Stefano Luzzatto +3
Mathematics · Physics and Astronomy · #37D25 #37G25 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS #msc:37D25 #msc:37G25

paper · pdf · doi:10.48550/arxiv.math/0306041

10 pages, 2 figures. This is an extended version of the paper "Hyperbolicity of periodic points for horseshoes with internal tangencies" by S. Luzzatto and I. Rios

openalex publication_date 2003/06/02 · arxiv created 2004/11/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly not uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.

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