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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
References:
[1]

S. Bautista and C. A. Morales, Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J., 6 (2006), 265-297, 406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. G. Bothe, How hyperbolic attractors determine their basins, Nonlinearity, 9 (1996), 1173-1190. doi: 10.1088/0951-7715/9/5/006.  Google Scholar

[5]

H. G. Bothe, Strange attractors with topologically simple basins, Topology Appl., 114 (2001), 1-25. doi: 10.1016/S0166-8641(00)00034-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.  Google Scholar

[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.  Google Scholar

[10]

J. Hempel, 3-Manifolds, Ann. of Math. Studies, No. 86, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1976.  Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

R. Metzger and C. A. Morales, Sectional-hyperbolic system, Ergod. Th. Dynam. Sys., 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.  Google Scholar

[13]

J. Milnor, Topology from the Differentiable Viewpoint, Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va., 1965.  Google Scholar

[14]

C. A. Morales, Sectional-Anosov flows, Monatsh. Math., 159 (2010), 253-260. doi: 10.1007/s00605-008-0078-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

S. Novikov, Topology of Foliations, Trudy Mosk. Mat. Obbsh., 14 (1965), 248-278 (AMS, translation 1967).  Google Scholar

[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.  Google Scholar

[20]

J. E. Reis, Infinidade de órbitas periódicas para fluxos seccional-Anosov, Tese (doutorado) - UFRJ/ IM/ Programa de Pós- Graduação em Matemática, 2011. Google Scholar

[21]

B. Scárdua and J. Seade, Codimension one foliations with Bott-Morse singularities. I, J. Differential Geom., 83 (2009), 189-212.  Google Scholar

[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.  Google Scholar

[23]

T. Vivier, Projective hyperbolicity and fixed points, Ergodic Theory Dynam. Systems, 26 (2006), 923-936. doi: 10.1017/S0143385705000581.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. G. Bothe, How hyperbolic attractors determine their basins, Nonlinearity, 9 (1996), 1173-1190. doi: 10.1088/0951-7715/9/5/006.  Google Scholar

[5]

H. G. Bothe, Strange attractors with topologically simple basins, Topology Appl., 114 (2001), 1-25. doi: 10.1016/S0166-8641(00)00034-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.  Google Scholar

[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.  Google Scholar

[10]

J. Hempel, 3-Manifolds, Ann. of Math. Studies, No. 86, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1976.  Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

R. Metzger and C. A. Morales, Sectional-hyperbolic system, Ergod. Th. Dynam. Sys., 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.  Google Scholar

[13]

J. Milnor, Topology from the Differentiable Viewpoint, Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va., 1965.  Google Scholar

[14]

C. A. Morales, Sectional-Anosov flows, Monatsh. Math., 159 (2010), 253-260. doi: 10.1007/s00605-008-0078-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

S. Novikov, Topology of Foliations, Trudy Mosk. Mat. Obbsh., 14 (1965), 248-278 (AMS, translation 1967).  Google Scholar

[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.  Google Scholar

[20]

J. E. Reis, Infinidade de órbitas periódicas para fluxos seccional-Anosov, Tese (doutorado) - UFRJ/ IM/ Programa de Pós- Graduação em Matemática, 2011. Google Scholar

[21]

B. Scárdua and J. Seade, Codimension one foliations with Bott-Morse singularities. I, J. Differential Geom., 83 (2009), 189-212.  Google Scholar

[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.  Google Scholar

[23]

T. Vivier, Projective hyperbolicity and fixed points, Ergodic Theory Dynam. Systems, 26 (2006), 923-936. doi: 10.1017/S0143385705000581.  Google Scholar

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