# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] A. Artigue, Lipschitz perturbations of expansive systems, Disc. Cont. Dyn. Syst., 35 (2015), 1829-1841. doi: 10.3934/dcds.2015.35.1829.  Google Scholar [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.  Google Scholar [4] A. Artigue, M. J. Pacífico and J. L. Vieitez, N-expansive homeomorphisms on surfaces,, Communications in Contemporary Mathematics, ().   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] 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.  Google Scholar [7] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. Li and R. Zhang, Levels of generalized expansiveness, preprint, arXiv:1503.03387, (2015). Google Scholar [11] R. Mañé, Expansive diffeomorphisms, in Dynamical Systems—Warwick 1974, Lecture Notes in Math., 468, Springer, Berlin, 1975, 162-174.  Google Scholar [12] C. A. Morales, Measure expansive systems, preprint, IMPA, 2011. Google Scholar [13] C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst., 32 (2012), 293-301. doi: 10.3934/dcds.2012.32.293.  Google Scholar [14] C. A. Morales and V. F. Sirvent, Expansive Measures, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. Palis and F. Takens, Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993.  Google Scholar [18] C. Robinson, $C^r$ structural stability implies Kupka-Smale, in Dynamical Systems (ed. Peixoto), Academic Press, New York, 1973, 443-449.  Google Scholar [19] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, 1995.  Google Scholar [20] K. Sakai, Continuum-wise expansive diffeomorphisms, Publicacions Matemátiques, 41 (1997), 375-382. doi: 10.5565/PUBLMAT_41297_04.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar [24] D. Yang and S. Gan, Expansive homoclinic classes, Nonlinearity, 22 (2009), 729-733. doi: 10.1088/0951-7715/22/4/002.  Google Scholar

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##### References:
 [1] A. Artigue, Kinematic expansive flows, Ergodic Theory and Dynamical Systems, available on CJO, (2014), 32pp. doi: 10.1017/etds.2014.65.  Google Scholar [2] A. Artigue, Lipschitz perturbations of expansive systems, Disc. Cont. Dyn. Syst., 35 (2015), 1829-1841. doi: 10.3934/dcds.2015.35.1829.  Google Scholar [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.  Google Scholar [4] A. Artigue, M. J. Pacífico and J. L. Vieitez, N-expansive homeomorphisms on surfaces,, Communications in Contemporary Mathematics, ().   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] 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.  Google Scholar [7] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. Li and R. Zhang, Levels of generalized expansiveness, preprint, arXiv:1503.03387, (2015). Google Scholar [11] R. Mañé, Expansive diffeomorphisms, in Dynamical Systems—Warwick 1974, Lecture Notes in Math., 468, Springer, Berlin, 1975, 162-174.  Google Scholar [12] C. A. Morales, Measure expansive systems, preprint, IMPA, 2011. Google Scholar [13] C. A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst., 32 (2012), 293-301. doi: 10.3934/dcds.2012.32.293.  Google Scholar [14] C. A. Morales and V. F. Sirvent, Expansive Measures, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. Palis and F. Takens, Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993.  Google Scholar [18] C. Robinson, $C^r$ structural stability implies Kupka-Smale, in Dynamical Systems (ed. Peixoto), Academic Press, New York, 1973, 443-449.  Google Scholar [19] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, 1995.  Google Scholar [20] K. Sakai, Continuum-wise expansive diffeomorphisms, Publicacions Matemátiques, 41 (1997), 375-382. doi: 10.5565/PUBLMAT_41297_04.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987. doi: 10.1007/978-1-4757-1947-5.  Google Scholar [24] D. Yang and S. Gan, Expansive homoclinic classes, Nonlinearity, 22 (2009), 729-733. doi: 10.1088/0951-7715/22/4/002.  Google Scholar
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