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

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  • Béguin François
    Laboratoire d'Analyse, Géométrie et Applications, Université Paris Nord
  • Boubaker Zouhour Rezig
    Universit du 7 Novembre à Carthage, Faculté des Sciences de Bizerte, Département de Mathématiques

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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.

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