American Institute of Mathematical Sciences

Advanced
April  2005, 12(3): 523-530. doi: 10.3934/dcds.2005.12.523

On $C^0$ genericity of various shadowing properties

Piotr Kościelniak 1, and  Marcin Mazur 1,

1. 

Uniwersytet Jagielloński, Instytut Matematyki, Reymonta 4, 30-059 Kraków, Poland, Poland

Received  July 2003 Revised  August 2004 Published  December 2004

We prove that the following properties are $C^0$ generic in the space of discrete dynamical systems on a compact smooth manifold $M$: periodic shadowing (Theorem 1.1) and, assuming dim$M\leq 3$, $\mathcal T_C$-inverse shadowing (Theorem 1.2).
Keywords: inverse) shadowing, (discrete) dynamical system, covering relations, handle decomposition, (periodic, Generic property.
Mathematics Subject Classification: Primary: 37C20, 37C50; Secondary: 54H2.
Citation: Piotr Kościelniak, Marcin Mazur. On $C^0$ genericity of various shadowing properties. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523
[1]

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
[2]

Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281
[3]

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

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

S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29
[6]

Mário Jorge Dias Carneiro, Alexandre Rocha. A generic property of exact magnetic Lagrangians. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4183-4194. doi: 10.3934/dcds.2012.32.4183
[7]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[8]

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

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641
[10]

Maciej J. Capiński. Covering relations and the existence of topologically normally hyperbolic invariant sets. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 705-725. doi: 10.3934/dcds.2009.23.705
[11]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[12]

Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113
[13]

Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091
[14]

Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621
[15]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[16]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963
[17]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259
[18]

Palle Jorgensen, Feng Tian. Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2307-2348. doi: 10.3934/dcdss.2019146
[19]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences