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

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.
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 & Continuous Dynamical Systems - A, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61
[1]

Boris Kalinin, Victoria Sadovskaya. Lyapunov exponents of cocycles over non-uniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5105-5118. doi: 10.3934/dcds.2018224

[2]

Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249

[3]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177

[4]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259

[5]

Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133

[6]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009

[7]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211

[8]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811

[9]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549

[10]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545

[11]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125

[12]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020

[13]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038

[14]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022

[15]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[16]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

[17]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233

[18]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[19]

Victor Magron, Marcelo Forets, Didier Henrion. Semidefinite approximations of invariant measures for polynomial systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019165

[20]

Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]