Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(6): 1421-1431

Online first article August 20, 2021      Printed November 1, 2021

https://doi.org/10.4134/JKMS.j210052

Copyright © The Korean Mathematical Society.

Preservation of expansivity in hyperspace dynamical systems

Namjip Koo, Hyunhee Lee

Chungnam National University; Chungnam National University

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 $C^0$-generically, any homeomorphism on a compact manifold is not hyper $N$-expansive for any $N\in \mathbb{N}$. Also we give some examples to illustrate our results.

Keywords: Hyperspace, expansiveness, $N$-expansiveness, continuum-wise expansiveness

MSC numbers: 37B05, 37C45, 37D20, 54H20, 54B20

Supported by: This work was supported by the National Research Foundations of Korea (NRF) grant funded by the Korea government (MSIT)(No. 2020R1F1A1A01068032).