Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1985-0800256-3
MathSciNet review: 800256
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $(M’, \phi ’)$ if and only if $M/\phi$ is order isomorphic to $M’/\phi ’$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F18

Retrieve articles in all journals with MSC: 58F18


Additional Information

Article copyright: © Copyright 1985 American Mathematical Society