A rescaled expansiveness for flows

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 ${\phi }_t$ 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 $d({\phi }_t(x),{\phi }_{\theta (t)}(y)\le \delta \Vert X({\phi }_t(x))\Vert$ for all $t\in \mathbb{R}$, then ${\phi }_{\theta (t)}(y)\in {\phi }_{[-\epsilon ,\epsilon ]}({\phi }_t(x))$ 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.

References in corpus (1)

• Topological entropy of Lorenz-like flows