Topological properties of sectional-Anosov flows

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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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (10) : 4735-4741. doi: 10.3934/dcds.2015.35.4735

References:

[1]	S. Bautista and C. A. Morales, Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J., 6 (2006), 265-297, 406.
[2]	J. S. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, 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.
[3]	C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888. doi: 10.1016/S0764-4442(97)80131-0.
[4]	H. G. Bothe, How hyperbolic attractors determine their basins, Nonlinearity, 9 (1996), 1173-1190. doi: 10.1088/0951-7715/9/5/006.
[5]	H. G. Bothe, Strange attractors with topologically simple basins, Topology Appl., 114 (2001), 1-25. doi: 10.1016/S0166-8641(00)00034-1.
[6]	C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5292-4.
[7]	J. Franks and B. Williams, Anomalous Anosov flows, in Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, 1980, 158-174.
[8]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
[9]	G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A and B, Vieweg, Braunschweig, 1986. doi: 10.1007/978-3-322-90115-6.
[10]	J. Hempel, 3-Manifolds, Ann. of Math. Studies, No. 86, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1976.
[11]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.
[12]	R. Metzger and C. A. Morales, Sectional-hyperbolic system, Ergod. Th. Dynam. Sys., 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.
[13]	J. Milnor, Topology from the Differentiable Viewpoint, Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va., 1965.
[14]	C. A. Morales, Sectional-Anosov flows, Monatsh. Math., 159 (2010), 253-260. doi: 10.1007/s00605-008-0078-7.
[15]	C. A. Morales, Incompressibility of tori transverse to Axiom A flows, Proc. Amer. Math. Soc., 136 (2008), 4349-4354. doi: 10.1090/S0002-9939-08-09409-4.
[16]	C. A. Morales, Singular-hyperbolic attractors with handlebody basins, J. Dyn. Control Syst., 13 (2007), 15-24. doi: 10.1007/s10883-006-9000-6.
[17]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.
[18]	S. Novikov, Topology of Foliations, Trudy Mosk. Mat. Obbsh., 14 (1965), 248-278 (AMS, translation 1967).
[19]	J. F. Plante and W. P. Thurston, Anosov flows and the fundamental group, Topology, 11 (1972), 147-150. doi: 10.1016/0040-9383(72)90002-X.
[20]	J. E. Reis, Infinidade de órbitas periódicas para fluxos seccional-Anosov, Tese (doutorado) - UFRJ/ IM/ Programa de Pós- Graduação em Matemática, 2011.
[21]	B. Scárdua and J. Seade, Codimension one foliations with Bott-Morse singularities. I, J. Differential Geom., 83 (2009), 189-212.
[22]	T. Sodero, Sectional-Anosov flows on certain compact 3-manifolds, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 439-454. doi: 10.1007/s00574-011-0024-5.
[23]	T. Vivier, Projective hyperbolicity and fixed points, Ergodic Theory Dynam. Systems, 26 (2006), 923-936. doi: 10.1017/S0143385705000581.

show all references

References:

[1]	S. Bautista and C. A. Morales, Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J., 6 (2006), 265-297, 406.
[2]	J. S. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, 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.
[3]	C. Bonatti, A. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 883-888. doi: 10.1016/S0764-4442(97)80131-0.
[4]	H. G. Bothe, How hyperbolic attractors determine their basins, Nonlinearity, 9 (1996), 1173-1190. doi: 10.1088/0951-7715/9/5/006.
[5]	H. G. Bothe, Strange attractors with topologically simple basins, Topology Appl., 114 (2001), 1-25. doi: 10.1016/S0166-8641(00)00034-1.
[6]	C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5292-4.
[7]	J. Franks and B. Williams, Anomalous Anosov flows, in Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), Lecture Notes in Math., 819, Springer, Berlin, 1980, 158-174.
[8]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
[9]	G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A and B, Vieweg, Braunschweig, 1986. doi: 10.1007/978-3-322-90115-6.
[10]	J. Hempel, 3-Manifolds, Ann. of Math. Studies, No. 86, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1976.
[11]	M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.
[12]	R. Metzger and C. A. Morales, Sectional-hyperbolic system, Ergod. Th. Dynam. Sys., 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.
[13]	J. Milnor, Topology from the Differentiable Viewpoint, Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va., 1965.
[14]	C. A. Morales, Sectional-Anosov flows, Monatsh. Math., 159 (2010), 253-260. doi: 10.1007/s00605-008-0078-7.
[15]	C. A. Morales, Incompressibility of tori transverse to Axiom A flows, Proc. Amer. Math. Soc., 136 (2008), 4349-4354. doi: 10.1090/S0002-9939-08-09409-4.
[16]	C. A. Morales, Singular-hyperbolic attractors with handlebody basins, J. Dyn. Control Syst., 13 (2007), 15-24. doi: 10.1007/s10883-006-9000-6.
[17]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.
[18]	S. Novikov, Topology of Foliations, Trudy Mosk. Mat. Obbsh., 14 (1965), 248-278 (AMS, translation 1967).
[19]	J. F. Plante and W. P. Thurston, Anosov flows and the fundamental group, Topology, 11 (1972), 147-150. doi: 10.1016/0040-9383(72)90002-X.
[20]	J. E. Reis, Infinidade de órbitas periódicas para fluxos seccional-Anosov, Tese (doutorado) - UFRJ/ IM/ Programa de Pós- Graduação em Matemática, 2011.
[21]	B. Scárdua and J. Seade, Codimension one foliations with Bott-Morse singularities. I, J. Differential Geom., 83 (2009), 189-212.
[22]	T. Sodero, Sectional-Anosov flows on certain compact 3-manifolds, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 439-454. doi: 10.1007/s00574-011-0024-5.
[23]	T. Vivier, Projective hyperbolicity and fixed points, Ergodic Theory Dynam. Systems, 26 (2006), 923-936. doi: 10.1017/S0143385705000581.

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