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Proceedings of the American Mathematical Society

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

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A topological characterization for non-wandering surface flows
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by Tomoo Yokoyama PDF
Proc. Amer. Math. Soc. 144 (2016), 315-323 Request permission

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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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
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 315-323
  • MSC (2010): Primary 37E35; Secondary 57R30
  • DOI: https://doi.org/10.1090/proc/12898
  • MathSciNet review: 3415598