Rotation Vectors for Surface Diffeomorphisms

We consider the concepts of rotation number and rotation vector for area preserving diffeomorphisms of surfaces and their applications. In the case that the surface is an annulus A the rotation number for a point x ∈ A represents an average rate at which the iterates of x rotate around the annulus. More generally the rotation vector takes values in the one-dimensional homology of the surface and represents the average “homological motion” of an orbit. There are two main results. The first is that if 0 is in the interior of the convex hull of the recurrent rotation vectors for an area preserving diffeomorphism ƒ isotopic to the identity, then ƒ has a fixed point of positive index. The second result asserts that if ƒ has a vanishing mean rotation vector, then ƒ has a fixed point of positive index. Applications include the result that an area preserving diffeomorphism of A that has at least one periodic point must in fact have infinitely many interior periodic points. This is a key step in the proof of the theorem that every smooth Riemannian metric on S2 has infinitely many distinct closed geodesies. Another application is a new proof of the Arnold conjecture for area preserving diffeomorphisms of closed oriented surfaces.