Equivalent definitions of rescaled expansiveness

Equivalent definitions of rescaled expansiveness

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.

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.