American Institute of Mathematical Sciences

Advanced
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 - A, 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.   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, (1981), 1.  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.  doi: 10.1016/S0764-4442(97)80131-0.  Google Scholar

[4]

H. G. Bothe, How hyperbolic attractors determine their basins,, Nonlinearity, 9 (1996), 1173.  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.  doi: 10.1016/S0166-8641(00)00034-1.  Google Scholar

[6]

C. Camacho and A. Lins Neto, Geometric Theory of Foliations,, Birkhäuser Boston, (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., (1979), 158.   Google Scholar

[8]

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

[9]

G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A and B,, Vieweg, (1986).  doi: 10.1007/978-3-322-90115-6.  Google Scholar

[10]

J. Hempel, 3-Manifolds,, Ann. of Math. Studies, (1976).   Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977).   Google Scholar

[12]

R. Metzger and C. A. Morales, Sectional-hyperbolic system,, Ergod. Th. Dynam. Sys., 28 (2008), 1587.  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, (1965).   Google Scholar

[14]

C. A. Morales, Sectional-Anosov flows,, Monatsh. Math., 159 (2010), 253.  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.  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.  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.  doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[18]

S. Novikov, Topology of Foliations,, Trudy Mosk. Mat. Obbsh., 14 (1965), 248.   Google Scholar

[19]

J. F. Plante and W. P. Thurston, Anosov flows and the fundamental group,, Topology, 11 (1972), 147.  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.   Google Scholar

[22]

T. Sodero, Sectional-Anosov flows on certain compact 3-manifolds,, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 439.  doi: 10.1007/s00574-011-0024-5.  Google Scholar

[23]

T. Vivier, Projective hyperbolicity and fixed points,, Ergodic Theory Dynam. Systems, 26 (2006), 923.  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.   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, (1981), 1.  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.  doi: 10.1016/S0764-4442(97)80131-0.  Google Scholar

[4]

H. G. Bothe, How hyperbolic attractors determine their basins,, Nonlinearity, 9 (1996), 1173.  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.  doi: 10.1016/S0166-8641(00)00034-1.  Google Scholar

[6]

C. Camacho and A. Lins Neto, Geometric Theory of Foliations,, Birkhäuser Boston, (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., (1979), 158.   Google Scholar

[8]

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

[9]

G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Parts A and B,, Vieweg, (1986).  doi: 10.1007/978-3-322-90115-6.  Google Scholar

[10]

J. Hempel, 3-Manifolds,, Ann. of Math. Studies, (1976).   Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Mathematics, (1977).   Google Scholar

[12]

R. Metzger and C. A. Morales, Sectional-hyperbolic system,, Ergod. Th. Dynam. Sys., 28 (2008), 1587.  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, (1965).   Google Scholar

[14]

C. A. Morales, Sectional-Anosov flows,, Monatsh. Math., 159 (2010), 253.  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.  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.  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.  doi: 10.1090/S0002-9939-99-04936-9.  Google Scholar

[18]

S. Novikov, Topology of Foliations,, Trudy Mosk. Mat. Obbsh., 14 (1965), 248.   Google Scholar

[19]

J. F. Plante and W. P. Thurston, Anosov flows and the fundamental group,, Topology, 11 (1972), 147.  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.   Google Scholar

[22]

T. Sodero, Sectional-Anosov flows on certain compact 3-manifolds,, Bull. Braz. Math. Soc. (N.S.), 42 (2011), 439.  doi: 10.1007/s00574-011-0024-5.  Google Scholar

[23]

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

[1]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[2]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[3]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[4]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[5]

M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403
[6]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[7]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[8]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[9]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[10]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[11]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[12]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[13]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[14]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[15]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075
[16]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[17]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[18]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[19]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[20]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences