Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

Abstract

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.