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February  2006, 15(1): 61-71. doi: 10.3934/dcds.2006.15.61

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

Yongluo Cao 1, , Stefano Luzzatto 2, and  Isabel Rios 3,

1. 

Department of Mathematics, Suzhou University, Suzhou 215006, Jiangsu, China
2. 

Dept. of Mathematics, Imperial College London, United Kingdom
3. 

Instituto de Matemática, Universidade Federal Fluminense, (UFF), Rio de Janeiro, Brazil

Received  November 2004 Revised  September 2005 Published  February 2006

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.
Keywords: invariant measures., non-hyperbolic systems, Horseshoes, Lyapunov exponents.
Mathematics Subject Classification: Primary: 37D25, Secondary: 37G2.
Citation: Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61
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