Geodesics on S² and periodic points of annulus homeomorphisms

We show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points. A consequence is the theorem that every smooth Riemannian metric on S² with positive Gaussian curvature has infinitely many distinct closed geodesics. In this paper we investigate area preserving homeomorphisms of the annulus and their periodic points. The main result is that an area preserving homeomorphism of the annulus which has at least one periodic point (perhaps on the boundary) must in fact have infinitely many interior periodic points. The motivation and main application of this result is the furthering of a program begun by Birkhoff [B] in his book "Dynamical Systems". There he shows that for many Riemannian metrics on S², including those with positive curvature, the problem of finding closed geodesics reduces to finding periodic points of a certain area preserving homeomorphism of the annulus. The annulus map in question can be shown to have a periodic point so our main result above can be applied to show the existence of infinitely many distinct closed geodesics whenever this annulus map exists. This is done in Sect. 4 Other quite different approaches to the problem of finding infinitely many geodesics have been successful in handling the cases which do not reduce to the investigation of an annulus homeomorphism (see [Ba]).