Controlled shadowing property

Alireza Zamani Bahabadi

Iran, Islamic Republic of

Ferdowsi University of Mashhad

Department of Pure Mathematics

Submitted: 2017-06-03

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Accepted: 2018-01-17

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Published: 2018-04-02

DOI: https://doi.org/10.4995/agt.2018.7731
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Keywords:

controlled shadowing property, chaos, topologically ergodic, specification property, topologically mixing

Supporting agencies:

This research was not funded

Abstract:

In this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specication properties. The relation between the controlled shadowing and chaos in sense of Li-Yorke is studied. At the end we give some examples to investigate the controlled shadowing property.
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References:

T. Arai and N. Chinen, P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy, Topology Appl. 154 (2007), 1254-1262. https://doi.org/10.1016/j.topol.2005.11.016

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. https://doi.org/10.1090/S0002-9947-1971-0274707-X

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc. 349 (1997), 2737-2754. https://doi.org/10.1090/S0002-9947-97-01873-4

B. A. Coomes, H. Koak and K. J. Palmer, Periodic shadowing, Chaotic numerics (Geelong,1993), in: Contemp. Math., Vol. 172, Amer. Math. Soc., Providence, RI, 1994, pp. 115-130.

R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl. 189 (1995), 409-423. https://doi.org/10.1006/jmaa.1995.1027

D. Ahmadi Dastjerdi and M. Hosseini, Sub-shadowings, Nonlinear Anal. 72 (2010), 3759-3766. https://doi.org/10.1016/j.na.2010.01.014

A. Fakhari and F. Helen Ghane, On shadowing: Ordinary and ergodic, J. Math. Anal. Appl. 364 (2010), 151-155. https://doi.org/10.1016/j.jmaa.2009.11.004

R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal. 67, no. 6 (2007), 1680-1689. https://doi.org/10.1016/j.na.2006.07.040

R. Gu, The average-shadowing property and topological ergodicity, J. Comput. Appl. Math. 206, no. 2 (2007), 796-800. https://doi.org/10.1016/j.cam.2006.08.025

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl. 117, no. 3 (2002), 259-272. https://doi.org/10.1016/S0166-8641(01)00025-6

P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms are bishadowing, Ann. Polon. Math. 65 (1997), 171-177. https://doi.org/10.4064/ap-65-2-171-177

D. Kwietniak and P. Oprocha, A note on the average shadowing property for expansive maps, Topology. Appl. 159 (2012), 19-27. https://doi.org/10.1016/j.topol.2011.04.016

M. Mazur, Tolerance stability conjecture revisited, Topology Appl. 131 (2003), 33-38. https://doi.org/10.1016/S0166-8641(02)00261-4

S. Yu. Pilyugin and O. B.Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999), 253-266. https://doi.org/10.1016/S0166-8641(98)00062-5

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in: Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 224-231. https://doi.org/10.1007/BFb0101795

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