Smooth Diffeomorphisms of Three-Dimensional Space with Stable Periodic Points

We consider a three-dimensional diffeomorphism with a hyperbolic point at the origin and a non-transversal homoclinic point to it. It is assumed that the stable manifold of the hyperbolic point is one-dimensional, while the unstable manifold is two-dimensional. It is assumed that the Jacobian matrix of the original diffeomorphism has a canonical form at the origin but is not diagonal. The tangency of the stable and unstable manifolds at the homoclinic point is not a tangency of finite order. It is shown that an arbitrary neighborhood of a homoclinic trajectory can contain an infinite set of one-pass stable periodic trajectories whose characteristic exponents are negative and bounded away from zero. L.P. Shilnikov, S. Newhouse and other authors investigated the neighborhood of a non-transversal homoclinic trajectory, assuming that the tangency of a stable and unstable manifold at a homoclinic point is a tangency of finite order. From their work it follows that under this condition, the neighborhood of a homoclinic trajectory does not contain one-pass stable periodic trajectories.