Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms

Abstract

The present paper concerns the dynamics of surface diffeomorphisms. Given a diffeomorphism f of a surface S, the torsion of the orbit of a point z ∈ S is, roughly speaking, the average speed of rotation of the tangent vectors under the action of the derivative of f, along the orbit of z under f. The purpose of the paper is to identify some situations where there exist measures and orbits with non-zero torsion. We prove that every area preserving diffeomorphism of the disc which coincides with the identity near the boundary has an orbit with non-zero torsion. We also prove that a diffeomorphism of the torus ${\mathbb T}^2$, isotopic to the identity, whose rotation set has non-empty interior, has an orbit with non-zero torsion.