Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms. (English) Zbl 1268.37064

Béguin, François; Boubaker, Zouhour Rezig

J. Math. Soc. Japan 65, No. 1, 137-168 (2013).

Summary: 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 \(z\in S\) 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.

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E45	Rotation numbers and vectors

Keywords:

torsion; Ruelle number; surface diffeomorphisms; rotation sets

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