# American Institute of Mathematical Sciences

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 and 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. [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). [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.

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##### References:
 [1] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp. [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). [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.
Classical geometric model of the Lorenz attractor.
Topological model of the Lorenz attractor.
The cylinder $B'$ is transverse to the flow.
The genus two surface S transverse to the flow and containing the Lorenz attractor.
A singular point of the stable foliation appears in $G$.
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