Figure 1.
Classical geometric model of the Lorenz attractor.
-
Previous Article
Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets
- DCDS Home
- This Issue
-
Next Article
The Chowla and the Sarnak conjectures from ergodic theory point of view
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 |
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.
Mathematics Subject Classification:
Primary: 37E35; Secondary: 54H20.
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. |
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. 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. |
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 $G$ .
| [1] |
Alfonso Artigue. Expansive flows of surfaces. Discrete and 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 and Continuous Dynamical Systems, 2022, 42 (3) : 1585-1598. doi: 10.3934/dcds.2021165 |
| [3] |
Luchezar Stoyanov. Pinching conditions, linearization and regularity of Axiom A flows. Discrete and 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 and 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 and 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 and Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178 |
| [8] |
Luis Barreira. Dimension theory of flows: A survey. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]