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October  2015, 35(10): 4735-4741. doi: 10.3934/dcds.2015.35.4735

Topological properties of sectional-Anosov flows

Enoch Humberto Apaza Calla 1, , Bulmer Mejia Garcia 1, and  Carlos Arnoldo Morales Rojas 2,

1. 

Departamento de Matemática, Universidade Federal de Viçosa, Viçosa-MG, Brazil, Brazil
2. 

Departamento de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil

Received  July 2014 Revised  February 2015 Published  April 2015

We study sectional-Anosov flows on compact $3$-manifolds. First we prove that every periodic orbits represents an infinite order element of the fundamental group outside the strong stable manifolds of the singularities. Next, in the transitive case, we prove that the first Betti number of the manifold is positive, that the number of singularities is given by the Euler characteristic and that every boundary's connected component has nonpositive Euler characteristic. Moreover, there is one component with negative characteristic if and only if the flow has singularities. These results will be used to discuss the existence of transitive sectional-Anosov flows on specific compact 3-manifolds with boundary.
Keywords: compact manifold, Sectional-Anosov flow, transitive flow, periodic orbit., handlebody.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37D5.
Citation: Enoch Humberto Apaza Calla, Bulmer Mejia Garcia, Carlos Arnoldo Morales Rojas. Topological properties of sectional-Anosov flows. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4735-4741. doi: 10.3934/dcds.2015.35.4735
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show all references

References:
[1]

Mosc. Math. J., 6 (2006), 265-297, 406.  Google Scholar

[2]

in Low-Dimensional Topology (San Francisco, Calif., 1981), Contemp. Math., 20, Amer. Math. Soc., Providence, RI, 1983, 1-60. doi: 10.1090/conm/020/718132.  Google Scholar

[3]

C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888. doi: 10.1016/S0764-4442(97)80131-0.  Google Scholar

[4]

Nonlinearity, 9 (1996), 1173-1190. doi: 10.1088/0951-7715/9/5/006.  Google Scholar

[5]

Topology Appl., 114 (2001), 1-25. doi: 10.1016/S0166-8641(00)00034-1.  Google Scholar

[6]

Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5292-4.  Google Scholar

[7]

in Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, 1980, 158-174.  Google Scholar

[8]

Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.  Google Scholar

[9]

Vieweg, Braunschweig, 1986. doi: 10.1007/978-3-322-90115-6.  Google Scholar

[10]

Ann. of Math. Studies, No. 86, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1976.  Google Scholar

[11]

Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

Ergod. Th. Dynam. Sys., 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.  Google Scholar

[13]

Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va., 1965.  Google Scholar

[14]

Monatsh. Math., 159 (2010), 253-260. doi: 10.1007/s00605-008-0078-7.  Google Scholar

[15]

Proc. Amer. Math. Soc., 136 (2008), 4349-4354. doi: 10.1090/S0002-9939-08-09409-4.  Google Scholar

[16]

J. Dyn. Control Syst., 13 (2007), 15-24. doi: 10.1007/s10883-006-9000-6.  Google Scholar

[17]

Proc. Amer. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[18]

Trudy Mosk. Mat. Obbsh., 14 (1965), 248-278 (AMS, translation 1967).  Google Scholar

[19]

Topology, 11 (1972), 147-150. doi: 10.1016/0040-9383(72)90002-X.  Google Scholar

[20]

Tese (doutorado) - UFRJ/ IM/ Programa de Pós- Graduação em Matemática, 2011. Google Scholar

[21]

J. Differential Geom., 83 (2009), 189-212.  Google Scholar

[22]

Bull. Braz. Math. Soc. (N.S.), 42 (2011), 439-454. doi: 10.1007/s00574-011-0024-5.  Google Scholar

[23]

Ergodic Theory Dynam. Systems, 26 (2006), 923-936. doi: 10.1017/S0143385705000581.  Google Scholar

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