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http://dx.doi.org/10.4134/CKMS.c180290

ON THE DENSITY OF VARIOUS SHADOWING PROPERTIES  

Koo, Namjip (Department of Mathematics Chungnam National University)
Tsegmid, Nyamdavaa (Department of Mathematics Chungnam National University)
Publication Information
Communications of the Korean Mathematical Society / v.34, no.3, 2019 , pp. 981-989 More about this Journal
Abstract
In this paper we deal with some shadowing properties of discrete dynamical systems on a compact metric space via the density of subdynamical systems. Let $f:X{\rightarrow}X$ be a continuous map of a compact metric space X and A be an f-invariant dense subspace of X. We show that if $f{\mid}_A:A{\rightarrow}A$ has the periodic shadowing property, then f has the periodic shadowing property. Also, we show that f has the finite average shadowing property if and only if $f{\mid}_A$ has the finite average shadowing property.
Keywords
periodic shadowing; finite average shadowing; invariant dense subset;
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1 D. V. Anosov, On a class of invariant sets of smooth dynamical systems, In: Proc. 5th Int. Conf. Nonl. Oscill., vol. 2, pp. 39-45, Kiev, 1970.
2 N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland Mathematical Library, 52, North-Holland Publishing Co., Amsterdam, 1994.
3 M. Baloush and S. C. Dzul-Kifli, The dynamics of 1-step shifts of finite type over two symbols, Indian J. Sci. Tech. 9 (2016), no. 46, 1-6. https://doi.org/10.17485/ijst/2016/v9i46/97733
4 A. D. Barwell, C. Good, P. Oprocha, and B. E. Raines, Characterizations of $\omega}$-limit sets in topologically hyperbolic systems, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 1819-1833. https://doi.org/10.3934/dcds.2013.33.1819   DOI
5 G. D. Birkhoff, An extension of Poincare's last geometric theorem, Acta Math. 47 (1926), no. 4, 297-311. https://doi.org/10.1007/BF02559515   DOI
6 M. L. Blank, Metric properties of $\epsilon$-trajectories of dynamical systems with stochastic behaviour, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 365-378. https://doi.org/10.1017/S014338570000451X   DOI
7 M. L. Blank, Small perturbations of chaotic dynamical systems, Russian Math. Surveys 44 (1989), no. 6, 1-33; translated from Uspekhi Mat. Nauk 44 (1989), no. 6(270), 3-28, 203. https://doi.org/10.1070/RM1989v044n06ABEH002302   DOI
8 C. Bonatti and S. Crovisier, Recurrence et genericite, Invent. Math. 158 (2004), no. 1, 33-104. https://doi.org/10.1007/s00222-004-0368-1   DOI
9 R. Bowen, $-\omega}$-limit sets for axiom A diffeomorphisms, J. Dierential Equations 18 (1975), no. 2, 333-339. https://doi.org/10.1016/0022-0396(75)90065-0   DOI
10 A. Darabi and A.-M. Forouzanfar, Periodic shadowing and standard shadowing property, Asian-Eur. J. Math. 10 (2017), no. 1, 1750006, 9 pp. https://doi.org/10.1142/S1793557117500061
11 L. Fernandez and C. Good, Shadowing for induced maps of hyperspaces, Fund. Math. 235 (2016), no. 3, 277-286. https://doi.org/10.4064/fm136-2-2016   DOI
12 P. Oprocha and X. Wu, On averaged tracing of periodic average pseudo orbits, Discrete Contin. Dyn. Syst. 37 (2017), no. 9, 4943-4957. https://doi.org/10.3934/dcds.2017212   DOI
13 C. Good and J. Meddaugh, Orbital shadowing, internal chain transitivity and $-\omega}$-limit sets, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 143-154. https://doi.org/10.1017/etds.2016.30   DOI
14 P. Koscielniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl. 310 (2005), no. 1, 188-196. https://doi.org/10.1016/j.jmaa.2005.01.053   DOI
15 P. Koscielniak and M. Mazur, On $C^0$ genericity of various shadowing properties, Discrete Contin. Dyn. Syst. 12 (2005), no. 3, 523-530. https://doi.org/10.3934/dcds.2005.12.523   DOI
16 M. Lee, Diffeomorphisms with periodic shadowing, Int. J. Math. Anal. (Ruse) 7 (2013), no. 38, 1895-1898. https://doi.org/10.12988/ijma.2013.35104   DOI
17 M. Lee, Notes on the eventual shadowing property of a continuous map, J. Chungcheong Math. Soc. 30 (2017), no. 4, 381-385. http://dx.doi.org/10.14403/jcms.2017.30.4.381   DOI
18 A. V. Osipov, S. Yu. Pilyugin, and S. B. Tikhomirov, Periodic shadowing and $\Omega$-stability, Regul. Chaotic Dyn. 15 (2010), no. 2-3, 404-417. https://doi.org/10.1134/S1560354710020255   DOI
19 K. Palmer, Shadowing in Dynamical Systems, Mathematics and its Applications, 501, Kluwer Academic Publishers, Dordrecht, 2000. https://doi.org/10.1007/978-1-4757-3210-8
20 S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math. 1706, Springer Verlag, Berlin, 1999. https://doi.org/10.1007/BFB0093184
21 C. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), no. 4, 1010-1021. https://doi.org/10.2307/2373414   DOI
22 S. Y. Pilyugin and K. Sakai, Shadowing and Hyperbolicity, Lecture Notes in Math. 2193, Springer International Publishing AG, 2017. https://doi.org/10.1007/978-3-319-65184-2