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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 $\mathrm {Per}(v)$ is open, there are no exceptional orbits, and that $\mathrm {P} \sqcup \mathrm {Sing}(v) = \{ x \in M \mid \omega (x) \cup \alpha (x) \subseteq \mathrm {Sing}(v) \}$, 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 $\overline {\mathrm {LD}\sqcup \mathrm {Per}(v)} \supseteq M - \mathrm {Sing}(v)$, 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 $\mathrm {int}(\mathrm {Per}(v) \sqcup \mathrm {Sing}(v)) = \emptyset$ and $M - (\mathrm {P} \sqcup \mathrm {Sing}(v))$ 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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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

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