Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit (Q2047543)

The author considers a two-dimensional diffeomorphism with a homoclinic orbit. It is assumed that the fixed point is hyperbolic, and the intersection of its stable and unstable manifolds is non-transversal at a homoclinic point. Sufficient conditions for which the tangency of the intersection is not of finite order are provided. For such cases, in the neighborhood of a non-transversal homoclinic point, there exist infinitely many \(r\)-pass stable periodic points with characteristic exponents bounded away from zero for \(r \in \{1, 2, 3\}\). These results extend previous ones, where the tangency of stable and unstable manifolds at a homoclinic point is of finite order.