Abstract: In this paper we consider two-dimensional diffeomorphisms with hyperbolic fixed points and nontransverse homoclinic points. It is assumed that the tangency of a stable and unstable manifolds is not a tangency of finite order. It is shown that there exists a continuous one-parameter set of two-dimensional diffeomorphisms such that each diffeomorphism in a neighborhood of a homoclinic point has an infinite set of stable periodic points whose characteristic exponents are separated from zero.