American Institute of Mathematical Sciences

Advanced
December  2009, 25(4): 1143-1162. doi: 10.3934/dcds.2009.25.1143

On the hyperbolicity of homoclinic classes

Christian Bonatti 1, , Shaobo Gan 2, and  Dawei Yang 3,

1. 

Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France
2. 

School of Mathematical Science, Peking University, Beijing 100871
3. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  October 2008 Revised  May 2009 Published  September 2009

We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class $H(p)$ admits a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where $E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all periodic points homoclinically related with $p$ are uniformly $E^u$-expanding at the period, then $H(p)$ is hyperbolic. We also give some consequences of this result.
Keywords: hyperbolic time, shadowing lemma., homoclinic class.
Mathematics Subject Classification: Primary: 37D20, 37D30; Secondary: 37D2.
Citation: Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143
[1]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[2]

Shaobo Gan. A generalized shadowing lemma. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 627-632. doi: 10.3934/dcds.2002.8.627
[3]

Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205
[4]

Xiao Wen, Lan Wen. No-shadowing for singular hyperbolic sets with a singularity. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 6043-6059. doi: 10.3934/dcds.2020258
[5]

Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107
[6]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[7]

Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787
[8]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1431-1445. doi: 10.3934/cpaa.2021027
[9]

S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761
[10]

Martín Sambarino, José L. Vieitez. Robustly expansive homoclinic classes are generically hyperbolic. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1325-1333. doi: 10.3934/dcds.2009.24.1325
[11]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[12]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128
[13]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[14]

Lorenzo J. Díaz, Jorge Rocha. How do hyperbolic homoclinic classes collide at heterodimensional cycles?. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 589-627. doi: 10.3934/dcds.2007.17.589
[15]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335
[16]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778
[17]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269
[18]

Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022
[19]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[20]

Todor Gramchev, Nicola Orrú. Cauchy problem for a class of nondiagonalizable hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 533-542. doi: 10.3934/proc.2011.2011.533

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (164)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy