Global 𝐶^{𝑟} structural stability of vector fields on open surfaces with finite genus

Abstract

A vector field X X on the open manifold M M is globally C r {C^r} structurally stable if X X has a neighborhood ∪ \cup in the Whitney C r {C^r} topology such that the trajectories of every vector field Y ∈ ∪ Y \in \cup can be mapped onto trajectories of X X by a homeomorphism h : M → M h:M \to M which is in a preassigned compact-open neighborhood of the identity. In [2] it was proved the theorem formulating the sufficient conditions for global C r ( r ≥ 1 ) {C^r}(r \geq 1) structural stability of vector fields on open surfaces ( dim ⁡ M = 2 ) (\dim M = 2) . These conditions are also necessary for global C r {C^r} structural stability on the plane if r ≥ 1 r \geq 1 (see [2]) and for r = 1 r = 1 on any open surface of finite genus [1]. Here we will generalize it for C r ( r ≥ 1 ) {C^r}(r \geq 1) vector fields defined on open orientable surface with finite genus and countable space of ends E E .