Document Zbl 1527.37027

On strong exponential limit shadowing property. (English) Zbl 1527.37027

Commun. Korean Math. Soc. 37, No. 4, 1249-1258 (2022).

Summary: In this study, we show that the strong exponential limit shadowing property (SELmSP, for short), which has been recently introduced, exists on a neighborhood of a hyperbolic set of a diffeomorphism. We also prove that \(\Omega\)-stable diffeomorphisms and \(\mathcal{L}\)-hyperbolic homeomorphisms have this type of shadowing property. By giving examples, it is shown that this type of shadowing is different from the other shadowings, and the chain transitivity and chain mixing are not necessary for it. Furthermore, we extend this type of shadowing property to positively expansive maps with the shadowing property.

MSC:

37C50	Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37D05	Dynamical systems with hyperbolic orbits and sets

Keywords:

exponential limit shadowing; limit shadowing; hyperbolic set; \(\Omega\)-stability; positively expansive map

Cite Review PDF

Full Text: DOI

References:

[1]	S. A. Ahmadi and M. R. Molaei, Exponential limit shadowing, Ann. Polon. Math. 108 (2013), no. 1, 1-10. https://doi.org/10.4064/ap108-1-1 · Zbl 1268.37022 · doi:10.4064/ap108-1-1
[2]	E. Akin, M. Hurley, and J. A. Kennedy, Dynamics of topologically generic homeomor-phisms, Mem. Amer. Math. Soc. 164 (2003), no. 783, viii+130 pp. https://doi.org/ 10.1090/memo/0783 · Zbl 1022.37010 · doi:10.1090/memo/0783
[3]	B. Carvalho and D. Kwietniak, On homeomorphisms with the two-sided limit shadowing property, J. Math. Anal. Appl. 420 (2014), no. 1, 801-813. https://doi.org/10.1016/ j.jmaa.2014.06.011 · Zbl 1310.37013 · doi:10.1016/j.jmaa.2014.06.011
[4]	T. Eirola, O. Nevanlinna, and S. Yu. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 75-92. https://doi.org/10.1080/01630569708816748 · Zbl 0881.58049 · doi:10.1080/01630569708816748
[5]	R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal. 67 (2007), no. 6, 1680-1689. https://doi.org/10.1016/j.na.2006.07.040 · Zbl 1121.37011 · doi:10.1016/j.na.2006.07.040
[6]	M. Kulczycki, D. Kwietniak, and P. Oprocha, On almost specification and average shad-owing properties, Fund. Math. 224 (2014), no. 3, 241-278. https://doi.org/10.4064/ fm224-3-4 · Zbl 1352.37068 · doi:10.4064/fm224-3-4
[7]	K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Con-tin. Dyn. Syst. 13 (2005), no. 2, 533-540. https://doi.org/10.3934/dcds.2005.13.533 · Zbl 1078.37015 · doi:10.3934/dcds.2005.13.533
[8]	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 · Zbl 0997.37001 · doi:10.1007/978-1-4757-3210-8
[9]	S. Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999. · Zbl 0954.37014
[10]	S. Yu. Pilyugin, Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations 19 (2007), no. 3, 747-775. https://doi.org/10.1007/ s10884-007-9073-2 · Zbl 1128.37018 · doi:10.1007/s10884-007-9073-2
[11]	S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity 23 (2010), no. 10, 2509-2515. https://doi.org/10.1088/0951-7715/23/ 10/009 · Zbl 1206.37012 · doi:10.1088/0951-7715/23/10/009
[12]	O. B. Plamenevskaya, Weak shadowing for two-dimensional diffeomorphisms, Mat. Za-metki 65 (1999), 477-480.
[13]	K. Sakai, Shadowing properties of L-hyperbolic homeomorphisms, Topology Appl. 112 (2001), no. 3, 229-243. https://doi.org/10.1016/S0166-8641(00)00002-X · Zbl 0983.37024 · doi:10.1016/S0166-8641(00)00002-X
[14]	K. Sakai, Various shadowing properties for positively expansive maps, Topology Appl. 131 (2003), no. 1, 15-31. https://doi.org/10.1016/S0166-8641(02)00260-2 · Zbl 1024.37016 · doi:10.1016/S0166-8641(02)00260-2

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.