Journal of the
Korean Mathematical Society
JKMS

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

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J. Korean Math. Soc. 2021; 58(1): 207-218

Online first article July 14, 2020      Printed January 1, 2021

https://doi.org/10.4134/JKMS.j200019

Copyright © The Korean Mathematical Society.

Positively expansive maps and the limit shadowing properties

Kazuhiro Sakai

Utsunomiya University

Abstract

In this paper, the notion of two-sided limit shadowing property is considered for a positively expansive open map. More precisely, let $f$ be a positively expansive open map of a compact metric space $X$. It is proved that if $f$ is topologically mixing, then it has the two-sided limit shadowing property. As a corollary, we have that if $X$ is connected, then the notions of the two-sided limit shadowing property and the average-shadowing property are equivalent.

Keywords: Positively expansive map, shadowing, limit shadowing, tow-sided limit shadowing, average shadowing, topologically transitive, topologically mixing

MSC numbers: Primary 37C50; Secondary 54H20, 37D20

Supported by: This work is supported by JSPS KAKENHI Grant Numbers 16K05167 and 19K03578