A rescaled expansiveness for flows

Abstract

We introduce a new version of expansiveness for flows. Let M M be a compact Riemannian manifold without boundary and let X X be a C 1 C^1 vector field on M M that generates a flow φ t \varphi _t on M M . We call X X rescaling expansive on a compact invariant set Λ \Lambda of X X if for any ϵ > 0 \epsilon >0 there is δ > 0 \delta >0 such that, for any x , y ∈ Λ x,y\in \Lambda and any time reparametrization θ : R → R \theta :\mathbb {R}\to \mathbb {R} , if d ( φ t ( x ) , φ θ ( t ) ( y ) ) ≤ δ ‖ X ( φ t ( x ) ) ‖ d(\varphi _t(x), \varphi _{\theta (t)}(y))\le \delta \|X(\varphi _t(x))\| for all t ∈ R t\in \mathbb R , then φ θ ( t ) ( y ) ∈ φ [ − ϵ , ϵ ] ( φ t ( x ) ) \varphi _{\theta (t)}(y)\in \varphi _{[-\epsilon , \epsilon ]}(\varphi _t(x)) for all t ∈ R 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.