Rescaled expansivity and separating flows

Abstract

In this article we give sufficient conditions for Komuro expansivity to imply the rescaled expansivity recently introduced by Wen and Wen. Also, we show that a flow on a compact metric space is expansive in the sense of Katok-Hasselblatt if and only if it is separating in the sense of Gura and the set of fixed points is open.Date: October 21, 2018.

Rescaled-Expansive Flows: Unstable Sets and Topological Entropy

“…Proof of Corollary E. Let φ be a k * -expansive flow such that Sing(φ) is a hyperbolic set. In [5] it is proved that φ is R-expansive. Now if Γ is an non-periodic attractor without singularities, then Theorem D implies directly the result.…”

“…This implies we only can have expansive flows with positive entropy in a higher dimensional setting. Another distinction is the existence of several distinct definitions of expansiveness for singular flows, such as k * -expansiveness, geometric expansiveness, kinematic expansiveness and rescaled-expansiveness (see [4] and [5] for details). This forces us to think carefully about what type of expansiveness is more appropriate to each context.…”

“…In [5], A. Artigue investigated the relationship between R-expansiveness and other forms of expansiveness for singular flows. In particular, it is obtained some criteria for k * -expasive flows to be R-expansive, as a consequence, we obtain the following corollary:…”

On Chaotic Behavior of ASH Attractors

“…On the other hand, R-expansiveness and K * -expansiveness are closely related properties for flows whose singularities exhibit nice structure. For precise information about these similarities we reefer the reader to the work [3], where it is studied the relation between R-expansiveness, K * -expansiveness and the separating property.…”

Spectral decomposition for rescaling expansive flows with rescaled shadowing

“…Recently, to study multisingular hyperbolicity of C 1 vector fields, Wen and Wen [14] introduced another notion of expansiveness as follows: an integrated flow φ of C 1 vector field F on compact smooth manifold M is called rescaling expansive on an invariant set Λ ⊂ M of φ if for 2268 WOOCHUL JUNG, NGOCTHACH NGUYEN AND YINONG YANG any ε > 0 there is δ > 0 such that for any x, y ∈ Λ and any continuous map h : R → R, if d(φ t (x), φ h(t) (y)) ≤ δ F (φ t (x)) for all t ∈ R, then φ h(t) (y) ∈ φ [−ε,ε] (φ t (x)) for all t ∈ R. Moreover, they proved that any multisingular hyperbolic system is rescaling expansive. The relationship among expansiveness, K * -expansiveness and rescaled expansiveness were studied in [3,14,15].…”