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

Singular cw-expansive flows

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.

Citation: Alfonso Artigue. Singular cw-expansive flows. Discrete & Continuous Dynamical Systems - A, 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.

[2]

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

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

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

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[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.

show all references

References:
[1]

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

[2]

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

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

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

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[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.

Figure 1.  Classical geometric model of the Lorenz attractor.
Figure 2.  Topological model of the Lorenz attractor.
Figure 3.  The cylinder $B'$ is transverse to the flow.
Figure 4.  The genus two surface S transverse to the flow and containing the Lorenz attractor.
Figure 5.  A singular point of the stable foliation appears in $G$.
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