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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Central sequences in flows on 2-manifolds of finite genus


Author: Dean A. Neumann
Journal: Proc. Amer. Math. Soc. 61 (1976), 39-43
MSC: Primary 58F99; Secondary 34C35
DOI: https://doi.org/10.1090/S0002-9939-1976-0426060-6
MathSciNet review: 0426060
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Abstract: Let $ \phi $ be a continuous flow on the metric space $ X$ and let $ {X^1},{X^2}, \ldots $ denote the ``central'' sequence of closed $ \phi $-invariant subsets of $ X$ obtained by iterating the process of taking nonwandering points of $ \phi $. A. Schwartz and E. Thomas have proved that, if $ X$ is an orientable $ 2$-manifold of finite genus, then this sequence can have not more than two distinct elements. We extend this result to include the nonorientable case; then this sequence can have at most three distinct elements. Analogous results are derived for the sequences obtained by iterating the processes of taking $ \alpha $ and $ \omega $ limit sets, or closures of $ \alpha $ and $ \omega $ limit sets.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0426060-6
Article copyright: © Copyright 1976 American Mathematical Society

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