Stability of periodic points of diffeomorphisms of multidimensional space

We study a diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a non-transversal homoclinic to it point. From the works of Sh. Newhouse, B. F. Ivanov, L. P. Shilnikov and other authors it follows that under a certain method of tangency of the stable and unstable manifolds, a neighborhood of a non-transversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of these points tends to zero with increasing period. In this paper, we study diffeomorphisms for which the method of tangency of the stable and unstable manifolds differs from the case studied in the works of the above-mentioned authors. This paper is a continuation of the author’s previous works, in which diffeomorphisms were studied, whose Jacobi matrices at the origin had only real eigenvalues. Conditions were obtained under which the neighborhood of a non-transversal homoclinic point of such diffeomorphism contains an infinite set of stable periodic points with characteristic separated from zero. In this paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has not only real eigenvalues, but also a non-unique pair of complex conjugate eigenvalues. Under this assumption, conditions are obtained for the presence in the neighborhood of a non-transversal homoclinic point of an infinite set of stable periodic points with characteristic exponents separated from zero.Работа выполнена при финансовой поддержке РФФИ (грант 16-01-00452)