A rescaled expansiveness for flows

TL;DR

This paper introduces a new concept called rescaling expansiveness for flows on manifolds, proving that multisingular hyperbolic sets exhibit this property and that the converse is generically true.

Contribution

The paper defines rescaling expansiveness for flows and establishes its equivalence with multisingular hyperbolic sets in a generic setting.

Findings

Multisingular hyperbolic sets are rescaling expansive.

Rescaling expansiveness is a new property for flows.

The converse of the main result holds generically.

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 $\varphi_t$ on $M$. We call $X$ {\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 $d(\varphi_t(x), \varphi_{\theta(t)}(y)\le \delta\|X(\varphi_t(x))\|$ 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 a converse holds generically.