Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical and Computer Sciences, Shahid Chamran University of Ahvaz, P.O. Box 61357-43135, Ahvaz, Iran.


In this paper, some properties of the periodic shadowing are presented. It is shown that a homeomorphism has the periodic shadowing property if and only if so does every lift of it to the universal covering space. Also, it is proved that continuous mappings on a compact metric space with the periodic shadowing and the average shadowing property also have the shadowing property and then are chaotic in the sense of Li-Yorke. Moreover, any distal homeomorphisms on a compact metric space with the periodic shadowing property do not have the asymptotic average shadowing property.


Main Subjects

[1] J. Blanks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), pp. 332-334.
[2] M.L. Blank, Small perturbations of chaotic dynamical systems, Russian Math. Surveys, 44 (1989), pp. 1-33.
[3] B. Carvalho, Two-sided limit shadowing Property, Ph.D. Thesis, Universidade Federal do Rio de Janeiro, 2015.
[4] A. Darabi and F. Forouzanfar, Periodic Shadowing and standard shadowing property, Asian-European J. Math., 10 (2017).
[5] R. Gu, The asymptotic average shadowing property and transitivity, J. Nonlinear Analysis., 67 (2007), pp. 1680-1689.
[6] K. Hiraide, Expansive homeomorphisms with the pseudo-orbit tracing property of n-tori, J. Math. Soc. Japan., 41 (1989), pp. 357-389.
[7] W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-York's chaos, Topology Appl., 117 (2002), pp. 259-272.
[8] A. Iwanik, Independent sets of transitive points, in Dynamical Systems and Ergodic Theory, Vol. 23 (Banach Center Publications, Warsaw, 1989), pp. 277-282.
[9] A. Koropecki and E. Pujals, Some consequences of the shadowing property in low dimensions, Ergodic Theory and Dynamical Systems., 195 (2013), pp. 1-37.
[10] P. Koscielniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl., 310 (2005), pp. 188-196.
[11] M. Kulczycki, D. Kwietniak, and P. Oprocha, On almost specification and average shadowing properties, Fund. Math., 224 (2014), pp. 241-278.
[12] A.V. Osipov, S.Yu. Pilyuginm, and S.B. Tikhomirov, Periodic shadowing and $Omega$-stability, Regular and Chaotic Dynamics, 15 (2010), pp. 404-417.
[13] K. Palmer, Shadowing in Dynamical Systems., Theory and Applications, Kluwer, Dordrecht, 2000.
[14] S.Yu. Pilyugin, Shadowing in Dynamical Systems, Lect. Notes in Math., Springer-Verlag, Berlin, 1999.
[15] O.B. Plamenevskaya, Weak shadowing for two-dimensional diffeomorphisms, Mat. Zametki., 65 (2013), pp. 477-480.