Bulletin of the
Korean Mathematical Society BKMS

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 $G_{\delta}$ 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: Periodic shadowable point, shadowable point, uniformly expansive point, dense subspace

MSC numbers: Primary 37B20, 37B65

Supported by: 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.