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

Abstract

Abstract.A vector field X on the open manifold M is globally C structurally stable if X has a neighborhood U in the Whitney C topology such that the trajectories of every vector field Y e U can be mapped onto trajectories of A' by a homeomorphism h: M -» M which is in a preassigned compact-open neighborhood of the identity. In The ideal boundary F of a surface M is the topological space having the equivalence classes of boundary components of M as elements and endowed with such a topology that E with it is homeomorphic to a subset of a Cantor set.By Hr(M) we denote the space of complete Cr vector fields on M with the Cr Whitney topology (r > 1). X, Y denote elements of Hr(M), 4>x denotes the flow induced by X. For x € M, Ox(x) (Ox(x) ,Ox(x)) is the trajectory of x (the positive semitrajectory, the negative semitrajectory) under 4>x . By Ox[x,y] we denote the closed X-trajectory segment from x to y .

On scattering trajectories of dynamical systems

“…18 Therefore L is foliated by R orbits and hence there is open scattering in D. 18 Therefore L is foliated by R orbits and hence there is open scattering in D.…”