## A topological characterization for non-wandering surface flows

HTML articles powered by AMS MathViewer

- 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 $\overline {\mathrm {P}} = \overline {\mathrm {LD}} =\mathbb {T}^2$.## References

- S. H. Aranson,
*Trajectories on nonorientable two-dimensional manifolds*, Mat. Sb. (N.S.)**80 (122)**(1969), 314–333 (Russian). MR**0259284** - S. Kh. Aranson, G. R. Belitsky, and E. V. Zhuzhoma,
*Introduction to the qualitative theory of dynamical systems on surfaces*, Translations of Mathematical Monographs, vol. 153, American Mathematical Society, Providence, RI, 1996. Translated from the Russian manuscript by H. H. McFaden. MR**1400885**, DOI 10.1090/mmono/153 - S. Aranson and E. Zhuzhoma,
*Maier’s theorems and geodesic laminations of surface flows*, J. Dynam. Control Systems**2**(1996), no. 4, 557–582. MR**1420359**, DOI 10.1007/BF02254703 - N. P. Bhatia and G. P. Szegö,
*Stability theory of dynamical systems*, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York-Berlin, 1970. MR**0289890** - T. M. Cherry,
*Topological Properties of the Solutions of Ordinary Differential Equations*, Amer. J. Math.**59**(1937), no. 4, 957–982. MR**1507295**, DOI 10.2307/2371361 - Milton Cobo, Carlos Gutierrez, and Jaume Llibre,
*Flows without wandering points on compact connected surfaces*, Trans. Amer. Math. Soc.**362**(2010), no. 9, 4569–4580. MR**2645042**, DOI 10.1090/S0002-9947-10-05113-5 - A. Denjoy,
*Sur les courbes définies par les équations differentielles a la surface du tore*J. Math. Pures Appl. (9) 11 (1932), 333–375. - John M. Franks,
*Two foliations in the plane*, Proc. Amer. Math. Soc.**58**(1976), 262–264. MR**415634**, DOI 10.1090/S0002-9939-1976-0415634-4 - Carlos Gutiérrez,
*Smoothing continuous flows on two-manifolds and recurrences*, Ergodic Theory Dynam. Systems**6**(1986), no. 1, 17–44. MR**837974**, DOI 10.1017/S0143385700003278 - Gilbert Hector and Ulrich Hirsch,
*Introduction to the geometry of foliations. Part A*, 2nd ed., Aspects of Mathematics, vol. 1, Friedr. Vieweg & Sohn, Braunschweig, 1986. Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy. MR**881799**, DOI 10.1007/978-3-322-90115-6 - Gilbert Hector,
*Quelques exemples de feuilletages espèces rares*, Ann. Inst. Fourier (Grenoble)**26**(1976), no. 1, xi, 239–264 (French, with English summary). MR**413128** - A. Mayer,
*Trajectories on the closed orientable surfaces*, Rec. Math. [Mat. Sbornik] N.S.**12(54)**(1943), 71–84 (Russian, with English summary). MR**0009485** - Nelson G. Markley,
*On the number of recurrent orbit closures*, Proc. Amer. Math. Soc.**25**(1970), 413–416. MR**256375**, DOI 10.1090/S0002-9939-1970-0256375-0 - Habib Marzougui,
*Flows with infinite singularities on closed two-manifolds*, J. Dynam. Control Systems**6**(2000), no. 4, 461–476. MR**1778209**, DOI 10.1023/A:1009574926153 - Habib Marzougui,
*Structure des feuilles sur les surfaces ouvertes*, C. R. Acad. Sci. Paris Sér. I Math.**323**(1996), no. 2, 185–188 (French, with English and French summaries). MR**1402540** - Habib Marzougui and Gabriel Soler López,
*Area preserving analytic flows with dense orbits*, Topology Appl.**156**(2009), no. 18, 3011–3015. MR**2556059**, DOI 10.1016/j.topol.2009.05.011 - Igor Nikolaev,
*Non-wandering flows on the 2-manifolds*, J. Differential Equations**173**(2001), no. 1, 1–16. MR**1836242**, DOI 10.1006/jdeq.2000.3924 - Igor Nikolaev and Evgeny Zhuzhoma,
*Flows on 2-dimensional manifolds*, Lecture Notes in Mathematics, vol. 1705, Springer-Verlag, Berlin, 1999. An overview. MR**1707298**, DOI 10.1007/BFb0093599 - H. Poincaré,
*Sur les courbes définies par une équation différentielle*IV, J. Math. Pures Appl. 85 (1886), 151–217. - J. H. Roberts and N. E. Steenrod,
*Monotone transformations of two-dimensional manifolds*, Ann. of Math. (2)**39**(1938), no. 4, 851–862. MR**1503441**, DOI 10.2307/1968468 - Arthur J. Schwartz,
*A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds*, Amer. J. Math. 85 (1963), 453-458; errata, ibid**85**(1963), 753. MR**0155061**

## 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