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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Global $ C\sp r$ structural stability of vector fields on open surfaces with finite genus


Author: Janina Kotus
Journal: Proc. Amer. Math. Soc. 108 (1990), 1039-1046
MSC: Primary 58F10; Secondary 34D30
MathSciNet review: 1004419
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1004419-3
PII: S 0002-9939(1990)1004419-3
Article copyright: © Copyright 1990 American Mathematical Society