May  2016, 36(5): 2367-2376. doi: 10.3934/dcds.2016.36.2367

Robustly N-expansive surface diffeomorphisms

1. 

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

Received  April 2015 Revised  May 2015 Published  October 2015

We give sufficient conditions for a diffeomorphism of a compact surface to be robustly $N$-expansive and cw-expansive in the $C^r$-topology. We give examples on the genus two surface showing that they need not to be Anosov diffeomorphisms. The examples are axiom A diffeomorphisms with tangencies at wandering points.
Citation: Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2367-2376. doi: 10.3934/dcds.2016.36.2367
References:
[1]

A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, available on CJO, (2014), 32pp. doi: 10.1017/etds.2014.65.

[2]

A. Artigue, Lipschitz perturbations of expansive systems, Disc. Cont. Dyn. Syst., 35 (2015), 1829-1841. doi: 10.3934/dcds.2015.35.1829.

[3]

A. Artigue and D. Carrasco-Olivera, A note on measure-expansive diffeomorphisms, J. Math. Anal. Appl., 428 (2015), 713-716. doi: 10.1016/j.jmaa.2015.02.052.

[4]

A. Artigue, M. J. Pacífico and J. L. Vieitez, N-expansive homeomorphisms on surfaces,, Communications in Contemporary Mathematics, (). 

[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]

J. Franks and C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. of the AMS, 223 (1976), 267-278. doi: 10.1090/S0002-9947-1976-0423420-9.

[7]

H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4.

[8]

H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology and its Applications, 53 (1993), 239-258. doi: 10.1016/0166-8641(93)90119-X.

[9]

M. Komuro, Expansive properties of Lorenz attractors, in The Theory of Dynamical Systems and Its Applications to Nonlinear Problems, World Sci. Publishing, Singapore, 1984, 4-26.

[10]

J. Li and R. Zhang, Levels of generalized expansiveness, preprint, arXiv:1503.03387, (2015).

[11]

R. Mañé, Expansive diffeomorphisms, in Dynamical Systems—Warwick 1974, Lecture Notes in Math., 468, Springer, Berlin, 1975, 162-174.

[12]

C. A. Morales, Measure expansive systems, preprint, IMPA, 2011.

[13]

C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst., 32 (2012), 293-301. doi: 10.3934/dcds.2012.32.293.

[14]

C. A. Morales and V. F. Sirvent, Expansive Measures, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013.

[15]

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.

[16]

R. Oliveira and F. Tari, On pairs of regular foliations in the plane, Cadernos de Matemática, 1 (2001), 167-180; Hokkaido Math. J., 31 (2002), 523-537. doi: 10.14492/hokmj/1350911901.

[17]

J. Palis and F. Takens, Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993.

[18]

C. Robinson, $C^r$ structural stability implies Kupka-Smale, in Dynamical Systems (ed. Peixoto), Academic Press, New York, 1973, 443-449.

[19]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, 1995.

[20]

K. Sakai, Continuum-wise expansive diffeomorphisms, Publicacions Matemátiques, 41 (1997), 375-382. doi: 10.5565/PUBLMAT_41297_04.

[21]

K. Sakai, N. Sumi and K. Yamamoto, Measure-expansive diffeomorphisms, J. Math. Anal. Appl., 414 (2014), 546-552. doi: 10.1016/j.jmaa.2014.01.023.

[22]

M. Sambarino and J. L. Vieitez, Robustly expansive homoclinic classes are generically hyperbolic, Discrete Contin. Dyn. Syst., 24 (2009), 1325-1333. doi: 10.3934/dcds.2009.24.1325.

[23]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987. doi: 10.1007/978-1-4757-1947-5.

[24]

D. Yang and S. Gan, Expansive homoclinic classes, Nonlinearity, 22 (2009), 729-733. doi: 10.1088/0951-7715/22/4/002.

show all references

References:
[1]

A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, available on CJO, (2014), 32pp. doi: 10.1017/etds.2014.65.

[2]

A. Artigue, Lipschitz perturbations of expansive systems, Disc. Cont. Dyn. Syst., 35 (2015), 1829-1841. doi: 10.3934/dcds.2015.35.1829.

[3]

A. Artigue and D. Carrasco-Olivera, A note on measure-expansive diffeomorphisms, J. Math. Anal. Appl., 428 (2015), 713-716. doi: 10.1016/j.jmaa.2015.02.052.

[4]

A. Artigue, M. J. Pacífico and J. L. Vieitez, N-expansive homeomorphisms on surfaces,, Communications in Contemporary Mathematics, (). 

[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]

J. Franks and C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. of the AMS, 223 (1976), 267-278. doi: 10.1090/S0002-9947-1976-0423420-9.

[7]

H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4.

[8]

H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology and its Applications, 53 (1993), 239-258. doi: 10.1016/0166-8641(93)90119-X.

[9]

M. Komuro, Expansive properties of Lorenz attractors, in The Theory of Dynamical Systems and Its Applications to Nonlinear Problems, World Sci. Publishing, Singapore, 1984, 4-26.

[10]

J. Li and R. Zhang, Levels of generalized expansiveness, preprint, arXiv:1503.03387, (2015).

[11]

R. Mañé, Expansive diffeomorphisms, in Dynamical Systems—Warwick 1974, Lecture Notes in Math., 468, Springer, Berlin, 1975, 162-174.

[12]

C. A. Morales, Measure expansive systems, preprint, IMPA, 2011.

[13]

C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst., 32 (2012), 293-301. doi: 10.3934/dcds.2012.32.293.

[14]

C. A. Morales and V. F. Sirvent, Expansive Measures, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013.

[15]

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.

[16]

R. Oliveira and F. Tari, On pairs of regular foliations in the plane, Cadernos de Matemática, 1 (2001), 167-180; Hokkaido Math. J., 31 (2002), 523-537. doi: 10.14492/hokmj/1350911901.

[17]

J. Palis and F. Takens, Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993.

[18]

C. Robinson, $C^r$ structural stability implies Kupka-Smale, in Dynamical Systems (ed. Peixoto), Academic Press, New York, 1973, 443-449.

[19]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, 1995.

[20]

K. Sakai, Continuum-wise expansive diffeomorphisms, Publicacions Matemátiques, 41 (1997), 375-382. doi: 10.5565/PUBLMAT_41297_04.

[21]

K. Sakai, N. Sumi and K. Yamamoto, Measure-expansive diffeomorphisms, J. Math. Anal. Appl., 414 (2014), 546-552. doi: 10.1016/j.jmaa.2014.01.023.

[22]

M. Sambarino and J. L. Vieitez, Robustly expansive homoclinic classes are generically hyperbolic, Discrete Contin. Dyn. Syst., 24 (2009), 1325-1333. doi: 10.3934/dcds.2009.24.1325.

[23]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987. doi: 10.1007/978-1-4757-1947-5.

[24]

D. Yang and S. Gan, Expansive homoclinic classes, Nonlinearity, 22 (2009), 729-733. doi: 10.1088/0951-7715/22/4/002.

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