  We prove that a "positive probability" subset of the boundary of the set of
hyperbolic (Axiom A) surface diffeomorphisms with no cycles $\mathcal{H}$ is
constituted by Kupka-Smale diffeomorphisms: all periodic points are hyperbolic
and their invariant manifolds intersect transversally. Lack of hyperbolicity
arises from the presence of a tangency between a stable manifold and an
unstable manifold, one of which is not associated to a periodic point. All
these diffeomorphisms that we construct lie on the boundary of the same
connected component of $\mathcal{H}$.
