A rescaled expansiveness for flows

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 ${\phi }_t$ on $M$. We call $X$ {\it rescaling expansive} on a compact invariant set $\mathrm{\Lambda }$ of $X$ if for any $ϵ>0$ there is $\delta >0$ such that, for any $x,y\in \mathrm{\Lambda }$ and any time reparametrization $\theta :\mathbb{R}\to \mathbb{R}$, if $d(\varphi_t(x), \varphi_{\theta(t)}(y)\le \delta|X(\varphi_t(x))|$forall$t\in \mathbb R$, then ${\phi }_{\theta (t)}(y)\in {\phi }_{[-ϵ,ϵ]}({\phi }_t(x))$ 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.