Multidimensional Diffeomorphisms with Stable Periodic Points

Abstract

Diffeomorphisms of a multidimensional space into itself with a hyperbolic fixed point are discussed in this paper. It is assumed that at the intersection of stable and unstable manifolds, there are points that are different from the hyperbolic point. Such points are called homoclinic and are divided into transversal and non-transversal, depending on the behavior of stable and unstable manifolds. It follows from articles by S. Newhouse, L. P. Shil’nikov, B. F. Ivanov, and others, that with a certain method of tangency of a stable manifold with an unstable one, the neighborhood of a non-transversal homoclinic point contains an infinite number of stable periodic points, but at least one of the characteristic exponents at these points tends to zero with increasing period. The present study is a continuation of previous studies by the author. In previously published papers, restrictions were imposed on the eigenvalues of the Jacobi matrix of the original diffeomorphism at a hyperbolic point. More precisely, it was assumed that either all eigenvalues are real and the Jacobi matrix is diagonal, or the matrix has only one real eigenvalue less than one in modulus, while all other eigenvalues are various complex integers greater than one in modulus. Within this framework, conditions are obtained for the presence of an infinite set of stable periodic points with characteristic exponents separated from zero in an arbitrary neighborhood of a non-transversal homoclinic point. It is assumed in this paper that the Jacobi matrix of a diffeomorphism has an arbitrary set of eigenvalues at a hyperbolic point. In this case, the conditions are obtained for the existence of an infinite set of stable periodic points whose characteristic exponents are separated from zero in the neighborhood of the non-transversal homoclinic point. The conditions are imposed, first of all, on the method of tangency of a stable manifold with an unstable one; however, the proof of the theorem essentially uses the properties of the eigenvalues of the Jacobi matrix at a hyperbolic point.