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

Kotus, Janina

doi:10.1090/S0002-9939-1990-1004419-3

Global $C^ r$ structural stability of vector fields on open surfaces with finite genus
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by Janina Kotus

Proc. Amer. Math. Soc. 108 (1990), 1039-1046

DOI: https://doi.org/10.1090/S0002-9939-1990-1004419-3

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Abstract:

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

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