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arXiv · 1706.09702

A rescaled expansiveness for flows

Xiao Wen·Lan Wen
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Abstract

We introduce a new version of expansiveness for flows. Let $M$ be a compact Riemannian manifold without boundary and $X$ be a $C^1$ vector field on $M$ that generates a flow $φ_t$ on $M$. We call $X$ {\it rescaling expansive} on a compact invariant set $Λ$ of $X$ if for any $ε>0$ there is $δ>0$ such that, for any $x,y\in Λ$ and any time reparametrization $θ:\mathbb{R}\to \mathbb{R}$, if $d(φ_t(x), φ_{θ(t)}(y)\le δ\|X(φ_t(x))\|$ for all $t\in \mathbb R$, then $φ_{θ(t)}(y)\in φ_{[-ε, ε]}(φ_t(x))$ for all $t\in \mathbb R$. We prove that every multisingular hyperbolic set (singular hyperbolic set in particular) is rescaling expansive and a converse holds generically.

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Xiao Wen, Lan Wen. 2017-06-29. A rescaled expansiveness for flows. https://arxiv.org/abs/1706.09702

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