American Institute of Mathematical Sciences

Advanced
August  2010, 27(3): 1133-1145. doi: 10.3934/dcds.2010.27.1133

Hyperbolicity of $C^1$-stably expansive homoclinic classes

Keonhee Lee 1, and  Manseob Lee 2,

1. 

Department of Mathematics, Chungnam National University, Daejeon, 305-764
2. 

Department of Mathematics, Mokwon University, Daejeon, 302-729, South Korea

Received  November 2008 Revised  February 2010 Published  March 2010

Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold, and let $p$ be a hyperbolic periodic point of $f$. In this paper we introduce the notion of $C^1$-stable expansivity for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $\mathcal {R}(f)$ of $f$ is $C^1$-stably expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, $(ii)$ the homoclinic class $H_f(p)$ of $f$ associated to $p$ is $C^1$-stably expansive if and only if $H_f(p)$ is hyperbolic, and $(iii)$ $C^1$-generically, the homoclinic class $H_f(p)$ is $C^1$-stably expansive if and only if $H_f(p)$ is $C^1$-persistently expansive.
Keywords: shadowing, chain component, homoclinic class, chain recurrent, germ expansive, Axiom A., $C^1$- persistently expansive, hyperbolic, $C^1$-stably expansive.
Mathematics Subject Classification: Primary: 37B20, 37C50; Secondary: 37D3.
Citation: 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
[1]

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
[2]

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
[3]

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
[4]

Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683
[5]

Flavio Abdenur, Lorenzo J. Díaz. Pseudo-orbit shadowing in the $C^1$ topology. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 223-245. doi: 10.3934/dcds.2007.17.223
[6]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1
[7]

S. Yu. Pilyugin, Kazuhiro Sakai, O. A. Tarakanov. Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 871-882. doi: 10.3934/dcds.2006.16.871
[8]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963
[9]

Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505
[10]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213
[11]

Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829
[12]

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
[13]

Alfonso Artigue. Singular cw-expansive flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126
[14]

Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252
[15]

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
[16]

Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307
[17]

Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789
[18]

Lan Wen. A uniform $C^1$ connecting lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 257-265. doi: 10.3934/dcds.2002.8.257
[19]

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
[20]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences