# American Institute of Mathematical Sciences

July  2011, 29(3): 693-736. doi: 10.3934/dcds.2011.29.693

## Simultaneous continuation of infinitely many sinks at homoclinic bifurcations

 1 Instituto de Matemática y Estadística Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay 2 Instituto de Matemática y Estadistica Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Uruguay 3 Instituto de Matemática y Estadistica Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay

Received  June 2009 Revised  August 2010 Published  November 2010

We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinks near the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.
Citation: Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693
##### References:
 [1] E. Colli, Infinitely many coexisting strange attractors,, Annales de l'I.H.P. Analyse non-linéaire, 15 (1998), 539. Google Scholar [2] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977). Google Scholar [3] W. de Melo, Structural stability of diffeomorphisms on two-manifolds,, Inventiones Math, 21 (1973), 233. doi: 10.1007/BF01390199. Google Scholar [4] A. Gorodetski and V. Kaloshin, How often surface diffeomorphisms have inifinitely sinks and hyperbolicity of periodic points near an homoclinic tangency,, Adv. Math., 208 (2007), 710. doi: 10.1016/j.aim.2006.03.012. Google Scholar [5] I. Kan, H. Koçak and J. A. Yorke, Antimonotonicity: Concurrent creation and annihilation of periodic orbits,, Ann. Math., 136 (1992), 219. doi: 10.2307/2946605. Google Scholar [6] S. Newhouse, Non density of Axiom A on $S^2$,, Proc. A.M.S. Symp. Pure Math., 14 (1970), 191. doi: 10.1016/0040-9383(74)90034-2. Google Scholar [7] S. Newhouse, Diffeomorphisms with infinitely many sinks,, Topology, 13 (1974), 9. Google Scholar [8] S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms,, Publ. IHÉS, 50 (1979), 101. Google Scholar [9] J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systèmes dynamiques (Orsay, 1995),, Astérisque, 261 (2000), 335. Google Scholar [10] J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynanics of Homoclinic Bifurcations,", University Press, (1993). Google Scholar [11] C. Robinson, Bifurcation to infinitely many sinks,, Comm Math Phys., 90 (1983), 433. doi: 10.1007/BF01206892. Google Scholar [12] M. Shub, "Global Stability of Dynamical Systems,", Springer Verlag, (1987), 23. Google Scholar [13] S. Smale, Diffeomorphisms with many periodic points,, in, (1965), 63. Google Scholar [14] L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?,, Comm. Math. Phys., 106 (1986), 635. doi: 10.1007/BF01463400. Google Scholar [15] J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes,, Bull AMS, 9 (1983), 319. doi: 10.1090/S0273-0979-1983-15191-1. Google Scholar

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##### References:
 [1] E. Colli, Infinitely many coexisting strange attractors,, Annales de l'I.H.P. Analyse non-linéaire, 15 (1998), 539. Google Scholar [2] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977). Google Scholar [3] W. de Melo, Structural stability of diffeomorphisms on two-manifolds,, Inventiones Math, 21 (1973), 233. doi: 10.1007/BF01390199. Google Scholar [4] A. Gorodetski and V. Kaloshin, How often surface diffeomorphisms have inifinitely sinks and hyperbolicity of periodic points near an homoclinic tangency,, Adv. Math., 208 (2007), 710. doi: 10.1016/j.aim.2006.03.012. Google Scholar [5] I. Kan, H. Koçak and J. A. Yorke, Antimonotonicity: Concurrent creation and annihilation of periodic orbits,, Ann. Math., 136 (1992), 219. doi: 10.2307/2946605. Google Scholar [6] S. Newhouse, Non density of Axiom A on $S^2$,, Proc. A.M.S. Symp. Pure Math., 14 (1970), 191. doi: 10.1016/0040-9383(74)90034-2. Google Scholar [7] S. Newhouse, Diffeomorphisms with infinitely many sinks,, Topology, 13 (1974), 9. Google Scholar [8] S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms,, Publ. IHÉS, 50 (1979), 101. Google Scholar [9] J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systèmes dynamiques (Orsay, 1995),, Astérisque, 261 (2000), 335. Google Scholar [10] J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynanics of Homoclinic Bifurcations,", University Press, (1993). Google Scholar [11] C. Robinson, Bifurcation to infinitely many sinks,, Comm Math Phys., 90 (1983), 433. doi: 10.1007/BF01206892. Google Scholar [12] M. Shub, "Global Stability of Dynamical Systems,", Springer Verlag, (1987), 23. Google Scholar [13] S. Smale, Diffeomorphisms with many periodic points,, in, (1965), 63. Google Scholar [14] L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?,, Comm. Math. Phys., 106 (1986), 635. doi: 10.1007/BF01463400. Google Scholar [15] J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes,, Bull AMS, 9 (1983), 319. doi: 10.1090/S0273-0979-1983-15191-1. Google Scholar
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