Journal of the
Korean Mathematical Society
JKMS

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

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J. Korean Math. Soc. 2018; 55(3): 593-604

Online first article December 19, 2017      Printed May 1, 2018

https://doi.org/10.4134/JKMS.j170326

Copyright © The Korean Mathematical Society.

Equivalent definitions of rescaled expansiveness

Xiao Wen, Yining Yu

Beihang University, Beihang University

Abstract

Recently, a new version of expansiveness which is closely attached to some certain weak version of hyperbolicity was given for $C^1$ vector fields as following: a $C^1$ vector field $X$ will be called {\it rescaling expansive} on a compact invariant set $\Lambda$ of $X$ if for any $\epsilon>0$ there is $\delta>0$ such that, for any $x,y\in \Lambda$ and any time reparametrization $\theta:\mathbb{R}\to \mathbb{R}$, if $d(\varphi_t(x), \varphi_{\theta(t)}(y))\leq \delta\|X(\varphi_t(x))\|$ for all $t\in \mathbb R$, then $\varphi_{\theta(t)}(y)\in \varphi_{(-\epsilon, \epsilon)}(\varphi_t(x))$ for all $t\in \mathbb R$. In this paper, some equivalent definitions for rescaled expansiveness are given.

Keywords: rescaling expansive, singular hyperbolic, multi-singular hyper\-bolic, flowbox

MSC numbers: Primary 37C10, 37D30

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