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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Completely unstable dynamical systems

Authors: Sudhir K. Goel and Dean A. Neumann
Journal: Trans. Amer. Math. Soc. 291 (1985), 639-668
MSC: Primary 58F18
MathSciNet review: 800256
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Abstract: We associate with the $ {C^r}\,(r\, \ge \,1)$ dynamical system $ \phi$ on an $ m$-manifold $ M$, the orbit space $ M/\phi$, defined to be the set of orbits of $ \phi$ with the quotient topology. If $ \phi$ is completely unstable, $ M/\phi$ turns out to be a $ {C^r}\,(m\, - \,1)$-nonseparated manifold. It is known that for a completely unstable flow $ \phi$ on a contractible manifold $ M,\,M/\phi$ is Hausdorff if and only if $ \phi$ is parallelizable. In general, we place an order on the non-Hausdorff points of $ M/\phi$ (essentially) by setting $ \bar p < \bar q $ if and only if $ {\pi ^{ - 1}}(\bar q) \subseteq {J^ + }({\pi ^{ - 1}}(\bar p))$. Our result is that $ (M,\,\phi)$ is topologically equivalent to $ (M^{\prime},\,\phi ^{\prime})$ if and only if $ M/\phi$ is order isomorphic to $ M^{\prime}/\phi ^{\prime}$.

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Article copyright: © Copyright 1985 American Mathematical Society