Equivalent definitions of rescaled expansiveness

Recently, a new version of expansiveness which is closely attached to some certain weak version of hyperbolicity was given for C¹ vector fields as following: a C¹ vector field X will be called rescaling expansive on a compact invariant set Λ of X if for any ɛ > 0 there is δ > 0 such that, for any x, y ∈ Λ and any time reparametrization θ: R → R, if d(ϕ_t(x), ϕ_θ(t) (y)) ≤ δ‖X(ϕ_t(x))‖ for all t ∈ R, then ϕ_θ(t) (y) ∈ϕ_(−ɛ,ɛ) (ϕ_t(x)) for all t ∈ R. In this paper, some equivalent definitions for rescaled expansiveness are given.