[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/CKMS.c200058

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)

Publication Information

Communications of the Korean Mathematical Society / v.35, no.4, 2020 , pp. 1309-1318 More about this Journal

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.

Keywords

Hyperspace; the induced map; topological stability; expansiveness; shadowing property;

Citations & Related Records

Reference

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 DOI
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 DOI
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 DOI
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 DOI
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 DOI
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 DOI
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 DOI
13	C. A. Morales, Shadowable points, Dyn. Syst. 31 (2016), no. 3, 347-356. https://doi.org/10.1080/14689367.2015.1131813 DOI
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.