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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
On the 75th birthday of Professor L.P. Shilnikov
Breaking hyperbolicity for smooth symplectic toral diffeomorphisms
L. Lerman 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
Аннотация:
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 $\pi_1$-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.
Ключевые слова:
symplectic diffeomorphism, torus, Anosov diffeomorphism, almost hyperbolic, degenerate saddle, bifurcation, homoclinic tangency, break of hyperbolicity, $\pi_1$-diffeomorphism.
Поступила в редакцию: 28.12.2009 Принята в печать: 23.02.2010
Образец цитирования:
L. Lerman, “Breaking hyperbolicity for smooth symplectic toral diffeomorphisms”, Regul. Chaotic Dyn., 15:2-3 (2010), 194–209
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd488 https://www.mathnet.ru/rus/rcd/v15/i2/p194
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