American Institute of Mathematical Sciences

Advanced
September  2008, 22(3): 683-697. doi: 10.3934/dcds.2008.22.683

$C^1$-stable shadowing diffeomorphisms

Keonhee Lee 1, , Kazumine Moriyasu 2, and  Kazuhiro Sakai 3,

1. 

Department of Mathematics, Chungnum National University, Daejeon 305-764, South Korea
2. 

Department of Mathematics, Tokushima University, Tokushima 770-8502, Japan
3. 

Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505

Received  July 2007 Revised  March 2008 Published  August 2008

Let $f$ be a diffeomorphism of a closed $C^\infty$ manifold. In this paper, we define the notion of the $C^1$-stable shadowing property for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $R(f)$ of $f$ has the $C^1$-stable shadowing property if and only if $f$ satisfies both Axiom A and the no-cycle condition, and $(ii)$ for the chain component $C_f(p)$ of $f$ containing a hyperbolic periodic point $p$, $C_f(p)$ has the $C^1$-stable shadowing property if and only if $C_f(p)$ is the hyperbolic homoclinic class of $p$.
Keywords: hyperbolic set, $C^1$-stable shadowing property, chain component, Axiom A., chain recurrent set, shadowing, pseudo-orbit, homoclinic class.
Mathematics Subject Classification: Primary: 37B20, 37C29, 37C50, 37D20, 37D30; Secondary: 37C7.
Citation: Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683
[1]

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

Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205
[3]

Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019
[4]

S. Yu. Pilyugin, Kazuhiro Sakai, O. A. Tarakanov. Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 871-882. doi: 10.3934/dcds.2006.16.871
[5]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963
[6]

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

Manseob Lee, Jumi Oh, Xiao Wen. Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1913-1927. doi: 10.3934/dcds.2020346
[8]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123
[9]

Jonathan Meddaugh. Shadowing as a structural property of the space of dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2439-2451. doi: 10.3934/dcds.2021197
[10]

Piotr Kościelniak, Marcin Mazur. On $C^0$ genericity of various shadowing properties. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523
[11]

Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307
[12]

Xiao Wen, Lan Wen. No-shadowing for singular hyperbolic sets with a singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6043-6059. doi: 10.3934/dcds.2020258
[13]

Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107
[14]

Víctor Ayala, Adriano Da Silva, Philippe Jouan. Jordan decomposition and the recurrent set of flows of automorphisms. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1543-1559. doi: 10.3934/dcds.2020330
[15]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040
[16]

Sergei Yu. Pilyugin. Variational shadowing. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733
[17]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[18]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[19]

Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, Katsutoshi Shinohara. How to identify a hyperbolic set as a blender. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6815-6836. doi: 10.3934/dcds.2020295
[20]

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

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy