2017/06/29 by Wen, Xiao, Wen, Lan · 3 citations
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.09702
We introduce a new version of expansiveness for flows. Let M be a compact Riemannian manifold without boundary and X be a C1 vector field on M that generates a flow φt 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∈ Λ and any time reparametrization θ:ℝ→ ℝ, if d(φt(x), φθ(t)(y)≤ δ‖X(φt(x))‖ for all t∈ \mathbb R, then φθ(t)(y)∈ φ[-ε, ε](φt(x)) for all t∈ \mathbb R. We prove that every multisingular hyperbolic set (singular hyperbolic set in particular) is rescaling expansive and a converse holds generically.