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

Rescaled expansivity and separating flows

Alfonso Artigue

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.

Keywords: Dynamical system, expansive, flow, Komuro expansivity, singular flow.
Mathematics Subject Classification: 37B05.
Citation: Alfonso Artigue. Rescaled expansivity and separating flows. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4433-4447. doi: 10.3934/dcds.2018193
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.  Google Scholar

[2]

A. Artigue, Expansive flows of surfaces, Disc. & Cont. Dyn. Sys., 33 (2013), 505-525.  doi: 10.3934/dcds.2013.33.505.  Google Scholar

[3]

A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, 36 (2016), 390-421.  doi: 10.1017/etds.2014.65.  Google Scholar

[4]

C. Bonatti and A. da Luz, Star Flows and Multisingular Hyperbolicity, arXiv, 2017. Google Scholar

[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.  Google Scholar

[6]

M. Brunella, Expansive flows on Seifert manifolds and torus bundles, Bol. Soc. Bras. Mat., 24 (1993), 89-104.   Google Scholar

[7]

W. Cordeiro, Fluxos CW-expansivos, Phd Thesis, UFRJ, Brazil, 2015. Google Scholar

[8]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1976.  Google Scholar

[9]

L. W. Flinn, Expansive Flows, Phd Thesis, University of Warwick, 1972. Google Scholar

[10]

A. A. Gura, Horocycle flow on a surface of negative curvature is separating, Mat. Zametki, 36 (1984), 279-284.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[14]

H. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory and Dynamical Systems, 1 (1981), 179-195.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199.   Google Scholar

[18]

X. Wen and L. Wen, A Rescaled Expansiveness for Flows, arXiv, 2017. Google Scholar

[19]

X. Wen and Y. Yu, Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., 55 (2018), 593-604.   Google Scholar

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.  Google Scholar

[2]

A. Artigue, Expansive flows of surfaces, Disc. & Cont. Dyn. Sys., 33 (2013), 505-525.  doi: 10.3934/dcds.2013.33.505.  Google Scholar

[3]

A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, 36 (2016), 390-421.  doi: 10.1017/etds.2014.65.  Google Scholar

[4]

C. Bonatti and A. da Luz, Star Flows and Multisingular Hyperbolicity, arXiv, 2017. Google Scholar

[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.  Google Scholar

[6]

M. Brunella, Expansive flows on Seifert manifolds and torus bundles, Bol. Soc. Bras. Mat., 24 (1993), 89-104.   Google Scholar

[7]

W. Cordeiro, Fluxos CW-expansivos, Phd Thesis, UFRJ, Brazil, 2015. Google Scholar

[8]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1976.  Google Scholar

[9]

L. W. Flinn, Expansive Flows, Phd Thesis, University of Warwick, 1972. Google Scholar

[10]

A. A. Gura, Horocycle flow on a surface of negative curvature is separating, Mat. Zametki, 36 (1984), 279-284.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[14]

H. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory and Dynamical Systems, 1 (1981), 179-195.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Paternain, Expansive flows and the fundamental group, Bull. Braz. Math. Soc., 24 (1993), 179-199.   Google Scholar

[18]

X. Wen and L. Wen, A Rescaled Expansiveness for Flows, arXiv, 2017. Google Scholar

[19]

X. Wen and Y. Yu, Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., 55 (2018), 593-604.   Google Scholar

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