Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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TOPOLOGICAL STABILITY IN HYPERSPACE DYNAMICAL SYSTEMS

  • Koo, Namjip (Department of Mathematics Chungnam National University) ;
  • Lee, Hyunhee (Department of Mathematics Chungnam National University) ;
  • Tsegmid, Nyamdavaa (Department of Mathematics Mongolian National University of Education)
  • Received : 2020.02.17
  • Accepted : 2020.07.20
  • Published : 2020.10.31
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Abstract

In this paper we extend the concept of topological stability from continuous maps to the corresponding induced maps and prove that a continuous map is topologically stable if and only if its induced map also is topologically stable.

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References

  1. D. V. Anosov, On a class of invariant sets of smooth dynamical systems, in Proc. 5th Int. Conf. Nonl. Oscill. Kiev. 2 (1970), 39-45.
  2. N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, 52, North-Holland Publishing Co., Amsterdam, 1994.
  3. A. Artigue, Hyper-expansive homeomorphisms, Publ. Mat. Urug. 14 (2013), 72-77.
  4. W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79 (1975), 81-92. https://doi.org/10.1007/BF01585664
  5. G. D. Birkhoff, An extension of Poincare's last geometric theorem, Acta Math. 47 (1926), no. 4, 297-311. https://doi.org/10.1007/BF02559515
  6. R. Bowen, ${\omega}$-limit sets for axiom A diffeomorphisms, J. Differential Equations 18 (1975), no. 2, 333-339. https://doi.org/10.1016/0022-0396(75)90065-0
  7. N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1047-1057. https://doi.org/10.1090/proc/13654
  8. A. Illanes and S. B. Nadler, Jr., Hyperspaces, Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York, 1999.
  9. I. S. Kim, H. Kato, and J. J. Park, On the countable compacta and expansive homeo-morphisms, Bull. Korean Math. Soc. 36 (1999), no. 2, 403-409.
  10. 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
  11. K. Lee, N.-T. Nguyen, and Y. Yang, Topological stability and spectral decomposition for homeomorphisms on noncompact spaces, Discrete Contin. Dyn. Syst. 38 (2018), no. 5, 2487-2503. https://doi.org/10.3934/dcds.2018103
  12. K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations 262 (2017), no. 6, 3467-3487. https://doi.org/10.1016/j.jde.2016.04.029
  13. C. A. Morales, Shadowable points, Dyn. Syst. 31 (2016), no. 3, 347-356. https://doi.org/10.1080/14689367.2015.1131813
  14. S. B. Nadler, Jr., Hyperspaces of Sets, Marcel Dekker, Inc., New York, 1978.
  15. S. Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999.
  16. S. Yu. Pilyugin and K. Sakai, Shadowing and hyperbolicity, Lecture Notes in Mathematics, 2193, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-65184-2
  17. W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. https://doi.org/10.2307/2031982
  18. P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 231-244, Lecture Notes in Math., 668, Springer, Berlin, 1978.
  19. Y. Wu and X. Xue, Shadowing property for induced set-valued dynamical systems of some expansive maps, Dynam. Systems Appl. 19 (2010), no. 3-4, 405-414.