Multi-Pass Stable Periodic Points of Diffeomorphism of a Plane with a Homoclinic Orbit

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a non-transversal point homoclinic to it is studied. There are various ways of touching stable and unstable manifolds at the homoclinic point. Periodic points whose trajectories do not leave the neighborhood of the trajectory of a homoclinic point are divided into many types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns lying outside a sufficiently small neighborhood of the hyperbolic point. Diffeomorphisms of the plane with a non-transversal homoclinic point were previously analyzed in the studies of Sh. Newhouse, L.P. Shil'nikov, and B.F. Ivanov, where it was assumed that this point is a tangency point of finite order. In these papers, it was shown that infinite sets of stable two-pass and three-pass periodic points can lie in a neighborhood of a homoclinic point. The presence of such sets depends on the properties of the hyperbolic point. In this paper, we assume that a homoclinic point is not a point with a finite-order tangency of a stable and an unstable manifold. It is shown that, for any fixed natural number n, the neighborhood of a non-transversal homoclinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents bounded away from zero.