Journal of the Mathematical Society of Japan

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

François BÉGUIN and Zouhour Rezig BOUBAKER

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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 $z\in S$ 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.

Article information

J. Math. Soc. Japan, Volume 65, Number 1 (2013), 137-168.

First available in Project Euclid: 24 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37E45: Rotation numbers and vectors

Torsion Ruelle number surface diffeomorphisms rotation sets


BÉGUIN, François; BOUBAKER, Zouhour Rezig. Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms. J. Math. Soc. Japan 65 (2013), no. 1, 137--168. doi:10.2969/jmsj/06510137.

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