Browse > Article
http://dx.doi.org/10.4134/JKMS.j210052

PRESERVATION OF EXPANSIVITY IN HYPERSPACE DYNAMICAL SYSTEMS  

Koo, Namjip (Department of Mathematics Chungnam National University)
Lee, Hyunhee (Department of Mathematics Chungnam National University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1421-1431 More about this Journal
Abstract
In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for n-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper N-expansive homeomorphisms via the topological dimension. More precisely, we show that C0-generically, any homeomorphism on a compact manifold is not hyper N-expansive for any N ∈ ℕ. Also we give some examples to illustrate our results.
Keywords
Hyperspace; expansiveness; N-expansiveness; continuum-wise expansiveness;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. T. Rogers, Jr., Dimension of hyperspaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20 (1972), 177-179.
2 H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), no. 3, 576-598. https://doi.org/10.4153/CJM-1993-030-4   DOI
3 J. Li and R. Zhang, Levels of generalized expansiveness, J. Dynam. Differential Equations 29 (2017), no. 3, 877-894. https://doi.org/10.1007/s10884-015-9502-6   DOI
4 R. Mane, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319. https://doi.org/10.2307/1998091   DOI
5 C. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 293-301. https://doi.org/10.3934/dcds.2012.32.293   DOI
6 Z. Nitecki, Differentialble Dynamics, MIT Press, 1971.
7 W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. https://doi.org/10.2307/2031982   DOI
8 A. Artigue, Kinematic expansive flows, Ergodic Theory Dynam. Systems 36 (2016), no. 2, 390-421. https://doi.org/10.1017/etds.2014.65   DOI
9 A. Artigue, Hyper-expansive homeomorphisms, Publ. Mat. Urug. 14 (2013), 72-77.
10 N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, 52, North-Holland Publishing Co., Amsterdam, 1994.
11 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
12 W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, NJ, 1941.
13 M. Levin and Y. Sternfeld, The space of subcontinua of a 2-dimensional continuum is infinite-dimensional, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2771-2775. https://doi.org/10.1090/S0002-9939-97-04172-5   DOI
14 N. C. Bernardes, Jr., and U. B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math. 231 (2012), no. 3-4, 1655-1680. https://doi.org/10.1016/j.aim.2012.05.024   DOI
15 S. Mazurkiewicz, Sur le type de dimension de l'hyperespace d'un continu, C. R. Soc. Sc. Varsovie 24 (1931), 191-192.
16 S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999), no. 3, 253-266. https://doi.org/10.1016/S0166-8641(98)00062-5   DOI
17 B. Carvalho and W. Cordeiro, n-expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (2016), no. 6, 3734-3755. https://doi.org/10.1016/j.jde.2016.06.003   DOI
18 A. Illanes and S. B. Nadler, Jr., Hyperspaces, Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York, 1999.