Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

ON STABILITY OF EXPANSIVE INDUCED HOMEOMORPHISMS ON HYPERSPACES

  • Koo, Namjip (Department of Mathematics Chungnam National University) ;
  • Lee, Hyunhee (Department of Mathematics Chungnam National University)
  • Received : 2022.01.07
  • Accepted : 2022.01.22
  • Published : 2022.02.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we investigate the topological stability of induced homeomorphisms on a hyperspace. More precisely, we show that an expansive induced homeomorphism on a hyperspace is topologically stable. We also give examples and a diagram about implications to illustrate our results.

Keywords

References

  1. N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, Elsevier science B. V., 1994.
  2. A. Artigue, Hyper-expansive homeomorphisms, Publ. Math. Del. Uruguay, 13 (2013), 72-76.
  3. N. P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057. https://doi.org/10.1090/proc/13654
  4. 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
  5. A. Illanes and Sam B. Nadler, Jr., Hyperspaces, Marcel Dekker Inc., New York and Basel, 1999.
  6. H. Kato and J.-J. Park, Expansive homeomorphisms of countable compata, Topology Appl., 95 (1999), 207-216. https://doi.org/10.1016/S0166-8641(98)00008-X
  7. N. Koo, K. Lee, and C. A. Morales, Pointwise topological stability, Proc. Edinb. Math. Soc. II, 61 (2018), no. 4, 1179-1191. https://doi.org/10.1017/S0013091518000263
  8. N. Koo, H. Lee and N. Tsegmid, Topological stability in hyperspace dynamical systems, Commun. Korean. Math. Soc., 35 (2020), no. 4, 1309-1318. https://doi.org/10.4134/CKMS.C200058
  9. N. Koo and G. Tumur, A note on weak expansive homeomorphisms on a compact metric space, J. Chungcheong Math. Soc., 34 (2020), no. 1, 95-101.
  10. K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487. https://doi.org/10.1016/j.jde.2016.04.029
  11. S. B. Nadler Jr., Hyperspaces of sets, Marcel Dekker Inc., New York and Basel, 1978.
  12. S. Y. Pilyugin, Shadowing in Dynamical Systems, Lect. Notes in Math., Springer-Verlag, Berlin, 1706, 1999.
  13. W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774. https://doi.org/10.1090/S0002-9939-1950-0038022-3
  14. P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The Structure of Attractors in Dynamical Systems, Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg and New York, 668 (1978), 231-244.