Uniform bounds for diffeomorphisms of the torus and a conjecture of Boyland

Abstract

We consider C 1 + ϵ diffeomorphisms of the torus, denoted by f , homotopic to the identity and whose rotation sets have non-empty interior. We give some uniform bounds on the displacement of points in the plane under iterates of a lift of f , relative to vectors in the boundary of the rotation set and we use these estimates in order to prove that if such a diffeomorphism f preserves area, then the rotation vector of the area measure is an interior point of the rotation set. This settles a strong version of a conjecture proposed by Philip Boyland. We also present some new results on the realization of extremal points of the rotation set by compact f-invariant subsets of the torus.