We show that for the standard map family, for all parameter values except one, the principal fixed point of the mapping has a transverse homoclinic point. As for the topological entropy, we show that it is positive for all parameter values. We also prove the following: Let S be a compact connected orientable surface and f an orientation preserving area preserving \(C ^1\) diffeomorphism of S. Suppose that U is an invariant domain of S such that its frontier in S has a finite number of connected components. Let b be a regular ideal boundary point of U which is fixed under the action induced by f on the ideal boundary of U, and let \({\hat{f}}\) be the orientation preserving homeomorphism induced on the corresponding circle of prime ends C(b). Let Z(b) be the impression of b in S and assume that all fixed points of f in Z(b) are non degenerate. If C(b) has a fixed prime end then C(b) has a finite number of fixed prime ends and there exists a semiconjugacy between the mapping of prime ends on C(b) and the restriction of f to Z(b). Furthermore, if p is the principal point of a fixed prime end then p is a fixed point of saddle type and Z(b) is a connected union of finitely many saddle connections and the corresponding saddles. In the case that U is homeomorphic to a disk and \(S \setminus U\) contains more than one point, this means that the frontier of U in S is a connected union of finitely many saddle connections and the corresponding saddles. This result can be seen as a two dimensional analogue of the dynamics of orientation preserving homeomorphisms of the circle with fixed points.
