Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PERIODIC SHADOWABLE POINTS

  • Namjip Koo (Department of Mathematics Chungnam National University) ;
  • Hyunhee Lee (Department of Mathematics Chungnam National University) ;
  • Nyamdavaa Tsegmid (Department of Mathematics Mongolian National University of Education)
  • Received : 2023.02.10
  • Accepted : 2023.09.22
  • Published : 2024.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider the set of periodic shadowable points for homeomorphisms of a compact metric space, and we prove that this set satisfies some properties such as invariance and being a Gδ set. Then we investigate implication relations related to sets consisting of shadowable points, periodic shadowable points and uniformly expansive points, respectively. Assume that the set of periodic points and the set of periodic shadowable points of a homeomorphism on a compact metric space are dense in X. Then we show that a homeomorphism has the periodic shadowing property if and only if so is the restricted map to the set of periodic shadowable points. We also give some examples related to our results.

Keywords

Acknowledgement

This work was supported by the National Research Foundations of Korea (NRF) grant funded by the Korea government(MSIT) (No.2020R1F1A1A01068032). The second author was supported by the National Research Foundations of Korea(NRF) grant funded by the Korea government(MSIT)(No.2021R1A6A3A13039168) and Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(RS-2023-00271567). The third author was supported by Mongolian National University of Education. The authors are grateful to the referees for the comments on the previous version of this paper.

References

  1. N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, 52, North-Holland, Amsterdam, 1994. 
  2. J. Aponte, Shadowable, topologically stable points and distal points for flows, PhD Thesis, Universidade Federal do Rio de Janeiro, UFRJ, 2017. 
  3. J. Aponte and H. Villavicencio, Shadowable points for flows, J. Dyn. Control Syst. 24 (2018), no. 4, 701-719. https://doi.org/10.1007/s10883-017-9381-8 
  4. A. Arbieto and E. F. Rego, Positive entropy through pointwise dynamics, Proc. Amer. Math. Soc. 148 (2020), no. 1, 263-271. https://doi.org/10.1090/proc/14682 
  5. 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 
  6. N. Kawaguchi, Quantitative shadowable points, Dyn. Syst. 32 (2017), no. 4, 504-518. https://doi.org/10.1080/14689367.2017.1280664 
  7. N. Koo and H. Lee, Topologically stable points and uniform limits, J. Korean Math. Soc. 60 (2023), no. 5, 1043-1055. https://doi.org/10.4134/JKMS.j220595 
  8. N. Koo, K. Lee, and C. A. Morales, Pointwise topological stability, Proc. Edinb. Math. Soc. (2) 61 (2018), no. 4, 1179-1191. https://doi.org/10.1017/s0013091518000263 
  9. N. Koo and N. Tsegmid, On the density of various shadowing properties, Commun. Korean Math. Soc. 34 (2019), no. 3, 981-989. https://doi.org/10.4134/CKMS.c180290 
  10. P. Koscielniak and M. Mazur, On C0 genericity of various shadowing properties, Discrete Contin. Dyn. Syst. 12 (2005), no. 3, 523-530. https://doi.org/10.3934/dcds.2005.12.523 
  11. C. A. Morales, Shadowable points, Dyn. Syst. 31 (2016), no. 3, 347-356. https://doi.org/10.1080/14689367.2015.1131813 
  12. S. Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, 1706, Springer, Berlin, 1999. 
  13. S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999), no. 3, 253-266. https://doi.org/10.1016/S0166-8641(98)00062-5