A rescaled expansiveness for flows

Wen, Xiao; Wen, Lan

doi:10.1090/tran/7382

A rescaled expansiveness for flows
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by Xiao Wen and Lan Wen PDF

Trans. Amer. Math. Soc. 371 (2019), 3179-3207 Request permission

Abstract:

We introduce a new version of expansiveness for flows. Let $M$ be a compact Riemannian manifold without boundary and let $X$ be a $C^1$ vector field on $M$ that generates a flow $\varphi _t$ on $M$. We call $X$ 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 $d(\varphi _t(x), \varphi _{\theta (t)}(y))\le \delta \|X(\varphi _t(x))\|$ for all $t\in \mathbb R$, then $\varphi _{\theta (t)}(y)\in \varphi _{[-\epsilon , \epsilon ]}(\varphi _t(x))$ for all $t\in \mathbb R$. We prove that every multisingular hyperbolic set (singular hyperbolic set in particular) is rescaling expansive and that a converse holds generically.

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