December 2018, 38(12): 5993-6013. doi: 10.3934/dcds.2018259

Robustly non-hyperbolic transitive symplectic dynamics

Instituto de Matemática e Estatística, UFF, Rua Prof. Marcos Waldemar de Freitas Reis, s/n, Niterói - RJ, 24210-201, Brazil

Received  July 2017 Revised  February 2018 Published  September 2018

We construct symplectomorphisms in dimension d ≥ 4 having a semi-local robustly transitive partially hyperbolic set containing C2-robust homoclinic tangencies of any codimension $c$ with 0 < cd/2.

Citation: Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259
References:
[1]

E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.

[2]

A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417, With an appendix by David Diica and Yakov Simpson-Weller. doi: 10.1017/S014338570700017X.

[3]

V. I. Arnold, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.

[4]

A. AvilaJ. Santamaria and M. Viana, Holonomy invariance: rough regularity and applications to lyapunov exponents, Astérisque, 358 (2013), 13-74.

[5]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189. doi: 10.1007/s00222-010-0243-1.

[6]

P. G. BarrientosY. Ki and A. Raibekas, Symbolic blender-horseshoes and applications, Nonlinearity, 27 (2014), 2805-2839. doi: 10.1088/0951-7715/27/12/2805.

[7]

P. G. Barrientos and A. Raibekas, Robust tangencies of large codimension, Nonlinearity, 30 (2017), 4369-4409. doi: 10.1088/1361-6544/aa8818.

[8]

C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and C1-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525. doi: 10.1017/S1474748008000030.

[9]

C. Bonatti and L. J. Díaz, Abundance of C1-homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148. doi: 10.1090/S0002-9947-2012-05445-6.

[10]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996), 357-396. doi: 10.2307/2118647.

[11]

J. BuzziS. Crovisier and T. Fisher, Local perturbations of conservative C1 diffeomorphisms, Nonlinearity, 30 (2017), 3613-3636. doi: 10.1088/1361-6544/aa803f.

[12]

O. Castejón, M. Guardia and V. Kaloshin, Random iteration of cylinder maps and diffusive behavior away from resonances, arXiv: 1705.09571.

[13]

R. de la Llave, Orbits of unbounded energy in perturbations of geodesic flow by potentials. a simple construction, 2002, Preprint.

[14]

A. DelshamsR. de la Llave and T. M. Seara, A geometric mechanism for diffusion in hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (2006), 1-141. doi: 10.1090/memo/0844.

[15]

P. Duarte, Abundance of elliptic islands at conservative bifurcations, Dynamics and Stability of Systems, 14 (1999), 339-356. doi: 10.1080/026811199281930.

[16]

P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergodic Theory Dynam. Systems, 20 (2000), 393-438. doi: 10.1017/S0143385700000195.

[17]

C.-W. Ho, On the periodic points of functions on a manifold, Proc. Amer. Math. Soc., 130 (2002), 2625-2630. doi: 10.1090/S0002-9939-02-06361-X.

[18]

A. J. Homburg and M. Nassiri, Robust minimality of iterated function systems with two generators, Ergodic Theory and Dynamical Systems, 34 (2014), 1914-1929. doi: 10.1017/etds.2013.34.

[19]

I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology, 17 (1978), 331-348. doi: 10.1016/0040-9383(78)90002-2.

[20]

J. M. Lee, Introduction to Smooth Manifolds, vol. 218 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2013.

[21]

J. Margalef-Roig and E. Dominguez, Differential Topology, North-Holland Mathematics Studies, Elsevier Science, 1992, https://books.google.cl/books?id=gexAr04vRT4C.

[22]

D. G. Schaeffer and M. Golubitsky, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5034-0.

[23]

L. Mora and N. Romero, Persistence of homoclinic tangencies for area-preserving maps, Annales de la Faculté des Sciences de Toulouse, 6 (1997), 711-725. doi: 10.5802/afst.885.

[24]

M. Nassiri and E. R. Pujals, Robust transitivity in hamiltonian dynamics, Ann. Sci. Éc. Norm. Supér., 45 (2012), 191-239. doi: 10.24033/asens.2164.

[25]

S. E. Newhouse, Nondensity of axiom A(a) on S2, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970,191-202. doi: 10.1007/978-3-642-16830-7.

[26]

J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms, J. Austral. Math. Soc. Ser. A, 61 (1996), 57-72. doi: 10.1017/S1446788700000070.

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.

[2]

A. Arbieto and C. Matheus, A pasting lemma and some applications for conservative systems, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417, With an appendix by David Diica and Yakov Simpson-Weller. doi: 10.1017/S014338570700017X.

[3]

V. I. Arnold, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.

[4]

A. AvilaJ. Santamaria and M. Viana, Holonomy invariance: rough regularity and applications to lyapunov exponents, Astérisque, 358 (2013), 13-74.

[5]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189. doi: 10.1007/s00222-010-0243-1.

[6]

P. G. BarrientosY. Ki and A. Raibekas, Symbolic blender-horseshoes and applications, Nonlinearity, 27 (2014), 2805-2839. doi: 10.1088/0951-7715/27/12/2805.

[7]

P. G. Barrientos and A. Raibekas, Robust tangencies of large codimension, Nonlinearity, 30 (2017), 4369-4409. doi: 10.1088/1361-6544/aa8818.

[8]

C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and C1-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525. doi: 10.1017/S1474748008000030.

[9]

C. Bonatti and L. J. Díaz, Abundance of C1-homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148. doi: 10.1090/S0002-9947-2012-05445-6.

[10]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996), 357-396. doi: 10.2307/2118647.

[11]

J. BuzziS. Crovisier and T. Fisher, Local perturbations of conservative C1 diffeomorphisms, Nonlinearity, 30 (2017), 3613-3636. doi: 10.1088/1361-6544/aa803f.

[12]

O. Castejón, M. Guardia and V. Kaloshin, Random iteration of cylinder maps and diffusive behavior away from resonances, arXiv: 1705.09571.

[13]

R. de la Llave, Orbits of unbounded energy in perturbations of geodesic flow by potentials. a simple construction, 2002, Preprint.

[14]

A. DelshamsR. de la Llave and T. M. Seara, A geometric mechanism for diffusion in hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (2006), 1-141. doi: 10.1090/memo/0844.

[15]

P. Duarte, Abundance of elliptic islands at conservative bifurcations, Dynamics and Stability of Systems, 14 (1999), 339-356. doi: 10.1080/026811199281930.

[16]

P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergodic Theory Dynam. Systems, 20 (2000), 393-438. doi: 10.1017/S0143385700000195.

[17]

C.-W. Ho, On the periodic points of functions on a manifold, Proc. Amer. Math. Soc., 130 (2002), 2625-2630. doi: 10.1090/S0002-9939-02-06361-X.

[18]

A. J. Homburg and M. Nassiri, Robust minimality of iterated function systems with two generators, Ergodic Theory and Dynamical Systems, 34 (2014), 1914-1929. doi: 10.1017/etds.2013.34.

[19]

I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology, 17 (1978), 331-348. doi: 10.1016/0040-9383(78)90002-2.

[20]

J. M. Lee, Introduction to Smooth Manifolds, vol. 218 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2013.

[21]

J. Margalef-Roig and E. Dominguez, Differential Topology, North-Holland Mathematics Studies, Elsevier Science, 1992, https://books.google.cl/books?id=gexAr04vRT4C.

[22]

D. G. Schaeffer and M. Golubitsky, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5034-0.

[23]

L. Mora and N. Romero, Persistence of homoclinic tangencies for area-preserving maps, Annales de la Faculté des Sciences de Toulouse, 6 (1997), 711-725. doi: 10.5802/afst.885.

[24]

M. Nassiri and E. R. Pujals, Robust transitivity in hamiltonian dynamics, Ann. Sci. Éc. Norm. Supér., 45 (2012), 191-239. doi: 10.24033/asens.2164.

[25]

S. E. Newhouse, Nondensity of axiom A(a) on S2, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970,191-202. doi: 10.1007/978-3-642-16830-7.

[26]

J. Ombach, Shadowing, expansiveness and hyperbolic homeomorphisms, J. Austral. Math. Soc. Ser. A, 61 (1996), 57-72. doi: 10.1017/S1446788700000070.

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