Korean J. Math. Vol. 32 No. 4 (2024) pp.639-646
DOI: https://doi.org/10.11568/kjm.2024.32.4.639

Egodic shadowable points and uniform limits

Main Article Content

Namjip Koo
Hyunhee Lee

Abstract

In this paper, we study some dynamical properties of ergodic shadowable points for dynamical systems on noncompact metric spaces. We also show that if a sequence of homeomorphisms on a metric space which converges uniformly to a homeomorphism has the ergodic shadowing property, then so does the uniform limit.



Article Details

Supporting Agencies

Chungnam National University

References

[1] S. A. Ahmadi, Shadowing, ergodic shadowing and uniform spaces, Filomat 31 (2017), no. 16, 5117–5124. https://dx.doi.org/10.2298/FIL1716117A Google Scholar

[2] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems: Recent Advances, North-Holland Mathematical Library, vol. 52., North-Holland Publishing Co., Amsterdam, 1994. Google Scholar

[3] J. Aponte and H. Villavicencio, Shadowable points for flows, J. Dyn. Control Syst. 24 (2018), no. 4, 701–719. https://dx.doi.org/10.1007/s10883-017-9381-8 Google Scholar

[4] P. K. Das and T. Das, Mean ergodic shadowing, Bull. Braz. Math. Soc. (N.S.) 54 (2023), no. 1, Paper no. 12, 12 pp. https://dx.doi.org/10.1007/s00574-022-00325-5 Google Scholar

[5] A. Fakhari and F. H. Ghane, On shadowing: ordinary and ergodic, J. Math. Anal. Appl. 364 (2010), 151–155. https://dx.doi.org/10.1016/j.jmaa.2009.11.004 Google Scholar

[6] N. Kawaguchi, Quantitative shadowable points, Dyn. Syst. 32 (2017), no. 4, 504–518. https://dx.doi.org/10.1080/14689367.2017.1280664 Google Scholar

[7] N. Kawaguchi, Properties of shadowable points: chaos and equicontinuity, Bull. Braz. Math. Soc. (N.S.) 48 (2017), no. 4, 599–622. https://dx.doi.org/10.1007/s00574-017-0033-0 Google Scholar

[8] N. Koo and H. Lee, Topologically stable points and uniform limits, J. Korean Math. Soc. 60 (2023), no. 5, 1043–1055. https://dx.doi.org/10.4134/JKMS.j220595 Google Scholar

[9] N. Koo and H. Lee, On the ergodic shadowing property through uniform limits, J. Chungcheong Math. Soc. 37 (2024), no. 2, 75–80. https://dx.doi.org/10.14403/jcms.2024.37.2.75 Google Scholar

[10] N. Koo and H. Lee, Totally ergodic shadowing property in dynamical systems on noncompact metric spaces, to appear in Bull. Korean Math. Soc. Google Scholar

[11] N. Koo, H. Lee, and N. Tsegmid, Periodic shadowable points, Bull. Korean Math. Soc. 61 (2024), no. 1, 195–205. https://dx.doi.org/10.4134/BKMS.b230071 Google Scholar

[12] N. Koo, K. Lee, and C. A. Morales, Pointwise topological stability, Proc. Edinb. Math. Soc. 61 (2018), no. 4, 1179–1191. https://dx.doi.org/10.1017/S0013091518000263 Google Scholar

[13] C. A. Morales, Shadowable points, Dyn Syst. 31 (2016), no. 3, 347–356. https://dx.doi.org/10.1080/14689367.2015.1131813 Google Scholar

[14] P. Oprocha, D. A. Dastjerdi, and M. Hosseini, On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl. 411 (2014), 454–463. https://dx.doi.org/10.1016/j.jmaa.2013.10.051 Google Scholar

[15] S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999. https://dx.doi.org/10.1007/BFb0093184 Google Scholar

[16] E. Rego, Uniform Limits and Pointwise Dynamics, PhD Thesis for Graduate School, UFRJ, 2017. Google Scholar

[17] E. Rego and A. Arbieto, Positive entropy through pointwise dynamics, Proc. Amer. Math. Soc. 148 (2020), no. 1, 263–271. https://dx.doi.org/10.1090/proc/14682 Google Scholar

[18] E. Rego and A. Arbieto, On the entropy of continuous flows with uniformly expansive points and the globalness of shadowable points with gaps, Bull. Braz. Math. Soc. (N.S.) 53 (2022), 853–872. https://dx.doi.org/10.1007/s00574-022-00285-w Google Scholar

[19] X. Wu, P. Oprocha, and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29 (2016), 1942–1972. https://dx.doi.org/10.1088/0951-7715/29/7/1942 Google Scholar

[20] Y. Yinong, Dynamical Systems on Noncompact Spaces, PhD Thesis for Graduate School, Chungnam National University, 2018. Google Scholar