September 2018, 38(9): 4433-4447. doi: 10.3934/dcds.2018193

Rescaled expansivity and separating flows

Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay

Received  August 2017 Published  June 2018

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.

Citation: Alfonso Artigue. Rescaled expansivity and separating flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4433-4447. doi: 10.3934/dcds.2018193
References:
[1]

V. AraujoM. J. PacificoE. 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. MoriyasuK. 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. AraujoM. J. PacificoE. 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. MoriyasuK. 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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