-
Previous Article
Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation
- DCDS Home
- This Issue
-
Next Article
The maximal entropy measure of Fatou boundaries
Rescaled expansivity and separating flows
Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay |
In this article we give sufficient conditions for Komuro expansivity to imply the rescaled expansivity recently introduced by Wen and Wen. Also, we show that a flow on a compact metric space is expansive in the sense of Katok-Hasselblatt if and only if it is separating in the sense of Gura and the set of fixed points is open.
References:
[1] |
V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana,
Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.
doi: 10.1090/S0002-9947-08-04595-9. |
[2] |
A. Artigue,
Expansive flows of surfaces, Disc. & Cont. Dyn. Sys., 33 (2013), 505-525.
doi: 10.3934/dcds.2013.33.505. |
[3] |
A. Artigue,
Kinematic expansive flows, Ergodic Theory and Dynamical Systems, 36 (2016), 390-421.
doi: 10.1017/etds.2014.65. |
[4] |
C. Bonatti and A. da Luz,
Star Flows and Multisingular Hyperbolicity, arXiv, 2017. |
[5] |
R. Bowen and P. Walters,
Expansive one-parameter flows, J. Diff. Eq., 12 (1972), 180-193.
doi: 10.1016/0022-0396(72)90013-7. |
[6] |
M. Brunella,
Expansive flows on Seifert manifolds and torus bundles, Bol. Soc. Bras. Mat., 24 (1993), 89-104.
|
[7] |
W. Cordeiro,
Fluxos CW-expansivos, Phd Thesis, UFRJ, Brazil, 2015. |
[8] |
M. P. do Carmo,
Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1976. |
[9] |
L. W. Flinn,
Expansive Flows, Phd Thesis, University of Warwick, 1972. |
[10] |
A. A. Gura,
Horocycle flow on a surface of negative curvature is separating, Mat. Zametki, 36 (1984), 279-284.
|
[11] |
U. Hamenstadt,
Dynamics of the Teichmuller flow on compact invariant sets, J. Mod. Dyn., 4 (2010), 393-418.
doi: 10.3934/jmd.2010.4.393. |
[12] |
T. Inaba and S. Matsumoto,
Nonsingular expansive flows on 3-manifolds and foliations with circle prong singularities, Japan. J. Math., 16 (1990), 329-340.
doi: 10.4099/math1924.16.329. |
[13] |
A. Katok and B. Hasselblatt,
Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511809187. |
[14] |
H. Keynes and M. Sears,
Real-expansive flows and topological dimension, Ergodic Theory and Dynamical Systems, 1 (1981), 179-195.
|
[15] |
M. Komuro, Expansive properties of Lorenz attractors, The Theory of Dynamical Systems and Its Applications to Nonlinear Problems, World Sci. Singapure, Kyoto, (1984), 4-26. |
[16] |
K. Moriyasu, K. Sakai and W. Sun,
$C^1$-stably expansive flows, Journal of Differential Equations, 213 (2005), 352-367.
doi: 10.1016/j.jde.2004.08.003. |
[17] |
M. Paternain,
Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199.
|
[18] |
X. Wen and L. Wen,
A Rescaled Expansiveness for Flows, arXiv, 2017. |
[19] |
X. Wen and Y. Yu,
Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., 55 (2018), 593-604.
|
show all references
References:
[1] |
V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana,
Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.
doi: 10.1090/S0002-9947-08-04595-9. |
[2] |
A. Artigue,
Expansive flows of surfaces, Disc. & Cont. Dyn. Sys., 33 (2013), 505-525.
doi: 10.3934/dcds.2013.33.505. |
[3] |
A. Artigue,
Kinematic expansive flows, Ergodic Theory and Dynamical Systems, 36 (2016), 390-421.
doi: 10.1017/etds.2014.65. |
[4] |
C. Bonatti and A. da Luz,
Star Flows and Multisingular Hyperbolicity, arXiv, 2017. |
[5] |
R. Bowen and P. Walters,
Expansive one-parameter flows, J. Diff. Eq., 12 (1972), 180-193.
doi: 10.1016/0022-0396(72)90013-7. |
[6] |
M. Brunella,
Expansive flows on Seifert manifolds and torus bundles, Bol. Soc. Bras. Mat., 24 (1993), 89-104.
|
[7] |
W. Cordeiro,
Fluxos CW-expansivos, Phd Thesis, UFRJ, Brazil, 2015. |
[8] |
M. P. do Carmo,
Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1976. |
[9] |
L. W. Flinn,
Expansive Flows, Phd Thesis, University of Warwick, 1972. |
[10] |
A. A. Gura,
Horocycle flow on a surface of negative curvature is separating, Mat. Zametki, 36 (1984), 279-284.
|
[11] |
U. Hamenstadt,
Dynamics of the Teichmuller flow on compact invariant sets, J. Mod. Dyn., 4 (2010), 393-418.
doi: 10.3934/jmd.2010.4.393. |
[12] |
T. Inaba and S. Matsumoto,
Nonsingular expansive flows on 3-manifolds and foliations with circle prong singularities, Japan. J. Math., 16 (1990), 329-340.
doi: 10.4099/math1924.16.329. |
[13] |
A. Katok and B. Hasselblatt,
Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511809187. |
[14] |
H. Keynes and M. Sears,
Real-expansive flows and topological dimension, Ergodic Theory and Dynamical Systems, 1 (1981), 179-195.
|
[15] |
M. Komuro, Expansive properties of Lorenz attractors, The Theory of Dynamical Systems and Its Applications to Nonlinear Problems, World Sci. Singapure, Kyoto, (1984), 4-26. |
[16] |
K. Moriyasu, K. Sakai and W. Sun,
$C^1$-stably expansive flows, Journal of Differential Equations, 213 (2005), 352-367.
doi: 10.1016/j.jde.2004.08.003. |
[17] |
M. Paternain,
Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199.
|
[18] |
X. Wen and L. Wen,
A Rescaled Expansiveness for Flows, arXiv, 2017. |
[19] |
X. Wen and Y. Yu,
Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., 55 (2018), 593-604.
|
[1] |
Mauricio Achigar. Extensions of expansive dynamical systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399 |
[2] |
Alfonso Artigue. Singular cw-expansive flows. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2945-2956. doi: 10.3934/dcds.2017126 |
[3] |
Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333 |
[4] |
Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239 |
[5] |
Benchawan Wiwatanapataphee, Yong Hong Wu, Thanongchai Siriapisith, Buraskorn Nuntadilok. Effect of branchings on blood flow in the system of human coronary arteries. Mathematical Biosciences & Engineering, 2012, 9 (1) : 199-214. doi: 10.3934/mbe.2012.9.199 |
[6] |
Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329 |
[7] |
Cheng-Zhong Xu, Gauthier Sallet. Multivariable boundary PI control and regulation of a fluid flow system. Mathematical Control and Related Fields, 2014, 4 (4) : 501-520. doi: 10.3934/mcrf.2014.4.501 |
[8] |
Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757 |
[9] |
Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks and Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749 |
[10] |
Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1 |
[11] |
Carlos Arnoldo Morales. A generalization of expansivity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 293-301. doi: 10.3934/dcds.2012.32.293 |
[12] |
Bernard Bonnard, Monique Chyba, Alain Jacquemard, John Marriott. Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields, 2013, 3 (4) : 397-432. doi: 10.3934/mcrf.2013.3.397 |
[13] |
G. C. Yang, K. Q. Lan. Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2465-2495. doi: 10.3934/cpaa.2013.12.2465 |
[14] |
Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93 |
[15] |
John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501 |
[16] |
Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson, Roger Temam. Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 509-546. doi: 10.3934/dcds.2018023 |
[17] |
Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035 |
[18] |
Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525 |
[19] |
Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks and Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41 |
[20] |
Magnus Aspenberg, Fredrik Ekström, Tomas Persson, Jörg Schmeling. On the asymptotics of the scenery flow. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2797-2815. doi: 10.3934/dcds.2015.35.2797 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]