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Breaking hyperbolicity for smooth symplectic toral diffeomorphisms

  • L.P. Shilnikov-75
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Abstract

We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on π 1-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.

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  1. Department of Differential Equations and Math. Analysis, and Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, Ulyanova ul. 10, Nizhny Novgorod, 603005, Russia

    L. Lerman

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Lerman, L. Breaking hyperbolicity for smooth symplectic toral diffeomorphisms. Regul. Chaot. Dyn. 15, 194–209 (2010). https://doi.org/10.1134/S1560354710020085

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