American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations
  • DCDS Home
  • This Issue
  • Next Article
    Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$
November  2009, 24(4): 1325-1333. doi: 10.3934/dcds.2009.24.1325

Robustly expansive homoclinic classes are generically hyperbolic

Martín Sambarino 1, and  José L. Vieitez 2,

1. 

Facultad de Ciencias, Iguá 4225 y Mataojo, Universidad de la República, Montevideo
2. 

Facultad deIngeniería, Herrera y Reissig 565, Universidad de la República, Montevideo, Uruguay

Received  July 2008 Revised  November 2008 Published  May 2009

Let $f: M \to M$ be a diffeomorphism defined in a $d$-dimensional compact boundary-less manifold $M$. We prove that generically $C^1$-robustly expansive homoclinic classes $H(p)$, $p$ an $f$-hyperbolic periodic point, are hyperbolic.
Keywords: Dominated Splitting, Hyperbolicity, Expansiveness, Homoclinic Classes..
Mathematics Subject Classification: Primary: 37D30; Secondary: 37D20,37C2.
Citation: Martín Sambarino, José L. Vieitez. Robustly expansive homoclinic classes are generically hyperbolic. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1325-1333. doi: 10.3934/dcds.2009.24.1325
[1]

Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143
[2]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[3]

Keonhee Lee, Manseob Lee. Hyperbolicity of $C^1$-stably expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1133-1145. doi: 10.3934/dcds.2010.27.1133
[4]

Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205
[5]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555
[6]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[7]

Xiao Wen. Structurally stable homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693
[8]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[9]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237
[10]

S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761
[11]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[12]

Martín Sambarino, José L. Vieitez. On $C^1$-persistently expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 465-481. doi: 10.3934/dcds.2006.14.465
[13]

Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179
[14]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335
[15]

Lorenzo J. Díaz, Jorge Rocha. How do hyperbolic homoclinic classes collide at heterodimensional cycles?. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 589-627. doi: 10.3934/dcds.2007.17.589
[16]

Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589
[17]

Katsutoshi Shinohara. On the index problem of $C^1$-generic wild homoclinic classes in dimension three. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 913-940. doi: 10.3934/dcds.2011.31.913
[18]

Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 585-602. doi: 10.3934/dcds.2003.9.585
[19]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757
[20]

Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences