Oriented Shadowing Property and $$\Omega $$ Ω -Stability for Vector Fields

Abstract

We call that a vector field has the oriented shadowing property if for any ε > 0 there is d > 0 such that each d-pseudo orbit is ε-oriented shadowed by some real orbit. In this paper, we show that the C 1 -interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the Ω-stability.2010 Mathematics Subject Classification. Primary 37C50.

An Example of a Vector Field with the Oriented Shadowing Property

“…Inequalities (3) imply that if δ is small enough, then dist(φ(t 3 + t, x 0 ), g(T p + t)) < ε/2, t ∈ [−T, 0].…”

“…Recently importance of this question appears again, during characterisation of vector fields with the C 1 -robust shadowing properties [3,6,13,14,18]. In particular in a work by the author [13] it was constructed an example of a not structurally stably vector field with the C 1 -robust oriented shadowing property.…”

Robustly shadowable chain transtive sets and hyperbolicity

“…To descirbe the dynamics on the underlying manifold, it is usual to use the dynamic properties on the tangent bundle such as hyperbolicity and dominated splitting. A fundamental problem in recent years is to study the influence of a robust dynamic property (i.e., property that holds for a given system and all C 1 -nearby systems) on the behavior of the tangent map on the tangent bundle (e.g., see [4,[6][7][8]10]).…”

“…Here we say that thewhere γ Y is the continuation of γ with respect to Y . Very recently, Gan et al [4] showed that the set of all robustly shadowable oriented vector fields is contained in the set of vector fields with Ω-stability. In this direction, the following question is still open: if the chain component C X (γ) of a C 1 -vector field X on a compact boundaryless Riemannian manifold M containing a hyperbolic periodic orbit γ is robustly shadowable, then is it hyperbolic?…”

Periodic shadowing of vector fields

“…They showed that a vector field satisfies the Lipschitz periodic shadowing property if and only if the vector field is Ω-stable. It becomes clear that the Lipschitz periodic shadowing property is strictly weaker than the standard shadowing property [4,10,13].…”