American Institute of Mathematical Sciences

Advanced
February  2005, 13(2): 533-540. doi: 10.3934/dcds.2005.13.533

Various shadowing properties and their equivalence

Keonhee Lee 1, and  Kazuhiro Sakai 2,

1. 

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

Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505

Received  June 2004 Revised  February 2005 Published  April 2005

In this paper, various shadowing properties are considered for expansive homeomorphisms. More precisely, we show that the continuous shadowing property, the Lipschitz shadowing property, the limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for expansive homeomorphisms on compact metric spaces.
Keywords: strong shadowing, limit shadowing, Pseudo-orbit, Lipschitz shadowing, continuous shadowing, expansive., shadowing.
Mathematics Subject Classification: Primary: 37B99, 37C15, 37C50, 37D20; Secondary: 54H2.
Citation: Keonhee Lee, Kazuhiro Sakai. Various shadowing properties and their equivalence. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 533-540. doi: 10.3934/dcds.2005.13.533
[1]

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
[2]

S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29
[3]

Sergei Yu. Pilyugin. Variational shadowing. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733
[4]

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

Davor Dragičević. Admissibility, a general type of Lipschitz shadowing and structural stability. Communications on Pure & Applied Analysis, 2015, 14 (3) : 861-880. doi: 10.3934/cpaa.2015.14.861
[6]

Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, Paweł Pilarczyk. Shadowing is generic---a continuous map case. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3591-3609. doi: 10.3934/dcds.2014.34.3591
[7]

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

Sergey V. Bolotin. Shadowing chains of collision orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 235-260. doi: 10.3934/dcds.2006.14.235
[9]

S. Yu. Pilyugin, A. A. Rodionova, Kazuhiro Sakai. Orbital and weak shadowing properties. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 287-308. doi: 10.3934/dcds.2003.9.287
[10]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
[11]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[12]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[13]

Will Brian, Jonathan Meddaugh, Brian Raines. Shadowing is generic on dendrites. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2211-2220. doi: 10.3934/dcdss.2019142
[14]

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
[15]

Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
[16]

Piotr Kościelniak, Marcin Mazur. On $C^0$ genericity of various shadowing properties. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523
[17]

Wolf-Jürgen Beyn, Sergey Piskarev. Shadowing for discrete approximations of abstract parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 19-42. doi: 10.3934/dcdsb.2008.10.19
[18]

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
[19]

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
[20]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences