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A topological characterization for non-wandering surface flows


Author: Tomoo Yokoyama
Journal: Proc. Amer. Math. Soc. 144 (2016), 315-323
MSC (2010): Primary 37E35; Secondary 57R30
DOI: https://doi.org/10.1090/proc/12898
Published electronically: September 15, 2015
MathSciNet review: 3415598
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Abstract: Let $ v$ be a continuous flow with arbitrary singularities on a compact surface. Then we show that if $ v$ is non-wandering, then $ v$ is topologically equivalent to a $ C^{\infty }$ flow such that $ \mathop {\mathrm {Per}}(v)$ is open, there are no exceptional orbits, and that $ \mathrm {P} \sqcup \mathop {\mathrm {Sing}}(v) = \{ x \in M \mid \omega (x) \cup \alpha (x) \subseteq \mathop {\mathrm {Sing}}(v) \}$, where $ \mathrm {P}$ is the union of non-closed proper orbits and $ \sqcup $ is the disjoint union symbol. Moreover, $ v$ is non-wandering if and only if $ \overline {\mathrm {LD}\sqcup \mathop {\mathrm {Per}}(v)} \supseteq M - \mathop {\mathrm {Sing}}(v)$, where $ \mathrm {LD}$ is the union of locally dense orbits and $ \overline {A}$ is the closure of a subset $ A \subseteq M$. On the other hand, $ v$ is topologically transitive if and only if $ v$ is non-wandering such that $ \mathop {\mathrm {int}}(\mathop {\mathrm {Per}}(v) \sqcup \mathop {\mathrm {Sing}}(v)) = \emptyset $ and $ M - (\mathrm {P} \sqcup \mathop {\mathrm {Sing}}(v))$ is connected, where $ \mathrm {int} {A}$ is the interior of a subset $ A \subseteq M$. In addition, we construct a smooth flow on $ \mathbb{T}^2$ with $ \overline {\mathrm {P}} = \overline {\mathrm {LD}} =\mathbb{T}^2$.


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Additional Information

Tomoo Yokoyama
Affiliation: Department of Mathematics, Faculty of Education, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522, Japan
Email: tomoo@kyokyo-u.ac.jp

DOI: https://doi.org/10.1090/proc/12898
Keywords: Surface flows, quasi-minimal sets
Received by editor(s): November 19, 2014
Received by editor(s) in revised form: November 21, 2014, and December 30, 2014
Published electronically: September 15, 2015
Additional Notes: The author was partially supported by the JST CREST Program at Department of Mathematics, Kyoto University of Education.
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

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