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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.
|
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