  We consider area preserving maps of surfaces and extend Mather's result on
the equality of the closure of the four branches of saddles. He assumed
elliptic fixed points to be Moser stable, while we require only that the
derivative at this points to be a rotation by an angle different from zero.
There are many results in the literature which require the hypothesis that
elliptic periodic points be Moser stable that now can be extended to the case
that the derivative at these points be an irrational rotation. The key point is
to give more information on Cartwright and Littlewood's fixed point theorem, to
show that the fixed point obtained by a fixed prime end can not be elliptic.
Hypotheses then became easier to verify: non degeneracy of fixed points and
nonexistence of saddle connections. As an application we show that the result
immediately implies that for the standard map family, for all values of the
parameter, except one, the principal hyperbolic fixed point has homoclinic
points. We also extend results to surfaces with boundary in order to be
applicable to return maps to surfaces of section and broken book
decompositions.
