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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Completely unstable dynamical systems
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by Sudhir K. Goel and Dean A. Neumann PDF
Trans. Amer. Math. Soc. 291 (1985), 639-668 Request permission

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 $M’/\phi ’$.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • 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