Rotation vectors and fixed points of area preserving surface diffeomorphisms

We consider the (homological) rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity. There are two main results. The first is that if O is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then f has a fixed point of positive index. The second result asserts that if f has a vanishing mean rotation vector then f has a fixed point of positive index. There are several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces.