American Institute of Mathematical Sciences

Advanced
June  2017, 37(6): 2945-2956. doi: 10.3934/dcds.2017126

Singular cw-expansive flows

Alfonso Artigue

Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay

Received  September 2016 Revised  January 2017 Published  February 2017

We study continuum-wise expansive flows with fixed points on metric spaces and low dimensional manifolds. We give sufficient conditions for a surface flow to be singular cw-expansive and examples showing that cw-expansivity does not imply expansivity. We also construct a singular Axiom A vector field on a three-manifold being singular cw-expansive and with a Lorenz attractor and a Lorenz repeller in its non-wandering set.

Keywords: Singular Axiom A, expansive flows, flows on surfaces, continuum theory, Lorenz attractor.
Mathematics Subject Classification: Primary: 37E35; Secondary: 54H20.
Citation: Alfonso Artigue. Singular cw-expansive flows. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126
References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp.   Google Scholar

[2]

S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, Translations of Mathematical Monographs AMS, 153 (1996).   Google Scholar

[3] V. Araújo and M. J. Pacífico, Three-Dimensional Flows, Springer-Verlag Berlin Heidelberg, 2010.  doi: 10.1007/978-3-642-11414-4.  Google Scholar
[4]

A. Artigue, Expansive flows of surfaces, Discrete and Continuous Dynamical Systems, 33 (2013), 505-525.  doi: 10.3934/dcds.2013.33.505.  Google Scholar

[5]

A. Artigue, Expansive flows of the three sphere, Differential Geometry and its Applications, 41 (2015), 91-101.  doi: 10.1016/j.difgeo.2015.04.006.  Google Scholar

[6]

A. Artigue, Robustly N-expansive surface diffeomorphisms, Discrete and Continuous Dynamical Systems, 36 (2016), 2367-2376.  doi: 10.3934/dcds.2016.36.2367.  Google Scholar

[7]

A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, 36 (2016), 390-421.  doi: 10.1017/etds.2014.65.  Google Scholar

[8]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[9]

W. Cordeiro, Fluxos CW-expansivos Thesis, UFRJ, Brazil, 2015. Google Scholar

[10]

L. W. Flinn, Expansive Flows University of Warwick, Thesis, 1972. Google Scholar

[11]

J. Franks and B. Williams, Anomalous Anosov Flows, Lecture Notes in Mathematics, 12 (1980), 158-174.   Google Scholar

[12]

L. F. He and G. Z. Shan, The nonexistence of expansive flow on a compact 2-manifold, Chinese Ann. Math. Ser. B, 12 (1991), 213-218.   Google Scholar

[13]

H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598.  doi: 10.4153/CJM-1993-030-4.  Google Scholar

[14]

M. Komuro, Expansive properties of Lorenz attractors, The theory of dynamical systems and its applications to nonlinear problems, (Kyoto, 1984), World Sci. Publishing, Singapore, 1984, 4-26.  Google Scholar

[15]

J. L. Massera, The meaning of stability, Bol. Fac. Ingen. Agrimens. Montevideo, 8 (1964), 405-429.   Google Scholar

[16]

C. A. Morales, A generalization of expansivity, Disc. and Cont. Dyn. Sys., 32 (2012), 293-301.  doi: 10.3934/dcds.2012.32.293.  Google Scholar

[17]

M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199.  doi: 10.1007/BF01237676.  Google Scholar

[18]

C. Robinson, Differentiability of the stable foliation for the model Lorenz equations, Lecture Notes in Mathematics, 898 (1981), 302-315.   Google Scholar

[19]

L. S. Young, Entropy of continuous flows on compact 2-manifolds, Topology, 16 (1977), 469-471.  doi: 10.1016/0040-9383(77)90053-2.  Google Scholar

show all references

References:
[1]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp.   Google Scholar

[2]

S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, Translations of Mathematical Monographs AMS, 153 (1996).   Google Scholar

[3] V. Araújo and M. J. Pacífico, Three-Dimensional Flows, Springer-Verlag Berlin Heidelberg, 2010.  doi: 10.1007/978-3-642-11414-4.  Google Scholar
[4]

A. Artigue, Expansive flows of surfaces, Discrete and Continuous Dynamical Systems, 33 (2013), 505-525.  doi: 10.3934/dcds.2013.33.505.  Google Scholar

[5]

A. Artigue, Expansive flows of the three sphere, Differential Geometry and its Applications, 41 (2015), 91-101.  doi: 10.1016/j.difgeo.2015.04.006.  Google Scholar

[6]

A. Artigue, Robustly N-expansive surface diffeomorphisms, Discrete and Continuous Dynamical Systems, 36 (2016), 2367-2376.  doi: 10.3934/dcds.2016.36.2367.  Google Scholar

[7]

A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, 36 (2016), 390-421.  doi: 10.1017/etds.2014.65.  Google Scholar

[8]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[9]

W. Cordeiro, Fluxos CW-expansivos Thesis, UFRJ, Brazil, 2015. Google Scholar

[10]

L. W. Flinn, Expansive Flows University of Warwick, Thesis, 1972. Google Scholar

[11]

J. Franks and B. Williams, Anomalous Anosov Flows, Lecture Notes in Mathematics, 12 (1980), 158-174.   Google Scholar

[12]

L. F. He and G. Z. Shan, The nonexistence of expansive flow on a compact 2-manifold, Chinese Ann. Math. Ser. B, 12 (1991), 213-218.   Google Scholar

[13]

H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598.  doi: 10.4153/CJM-1993-030-4.  Google Scholar

[14]

M. Komuro, Expansive properties of Lorenz attractors, The theory of dynamical systems and its applications to nonlinear problems, (Kyoto, 1984), World Sci. Publishing, Singapore, 1984, 4-26.  Google Scholar

[15]

J. L. Massera, The meaning of stability, Bol. Fac. Ingen. Agrimens. Montevideo, 8 (1964), 405-429.   Google Scholar

[16]

C. A. Morales, A generalization of expansivity, Disc. and Cont. Dyn. Sys., 32 (2012), 293-301.  doi: 10.3934/dcds.2012.32.293.  Google Scholar

[17]

M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199.  doi: 10.1007/BF01237676.  Google Scholar

[18]

C. Robinson, Differentiability of the stable foliation for the model Lorenz equations, Lecture Notes in Mathematics, 898 (1981), 302-315.   Google Scholar

[19]

L. S. Young, Entropy of continuous flows on compact 2-manifolds, Topology, 16 (1977), 469-471.  doi: 10.1016/0040-9383(77)90053-2.  Google Scholar

Figure 1.  Classical geometric model of the Lorenz attractor.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Topological model of the Lorenz attractor.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The cylinder $B'$ is transverse to the flow.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The genus two surface S transverse to the flow and containing the Lorenz attractor.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  A singular point of the stable foliation appears in $G$.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505
[2]

Se-Hyun Ku. Expansive flows on uniform spaces. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021165
[3]

Luchezar Stoyanov. Pinching conditions, linearization and regularity of Axiom A flows. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 391-412. doi: 10.3934/dcds.2013.33.391
[4]

Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1
[5]

Shin Kiriki, Ming-Chia Li, Teruhiko Soma. Geometric Lorenz flows with historic behavior. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7021-7028. doi: 10.3934/dcds.2016105
[6]

Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113
[7]

José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178
[8]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[9]

Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[10]

Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059
[11]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403
[12]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[13]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13
[14]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[15]

Yi Shi, Shaobo Gan, Lan Wen. On the singular-hyperbolicity of star flows. Journal of Modern Dynamics, 2014, 8 (2) : 191-219. doi: 10.3934/jmd.2014.8.191
[16]

Xiaoming Wang. On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 489-501. doi: 10.3934/dcdsb.2010.13.489
[17]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657
[18]

Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[19]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[20]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (125)
  • HTML views (58)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy