Torsion and Linking number for a surface diffeomorphism

Abstract

For a ${\mathcal{C}}^1$ diffeomorphism $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ isotopic to the identity, we prove that for any value $l\in \mathbb{R}$ of the linking number at finite time of the orbits of two points there exists at least a point whose torsion at the same finite time equals $l\in \mathbb{R}$. As an outcome, we give a much simpler proof of a theorem by Matsumoto and Nakayama concerning torsion of measure on ${\mathbb{T}}^2$. In addition, in the framework of twist maps, we generalize a known result concerning the linking number of periodic points: indeed, we estimate such value for any couple of points for which the limit of the linking number exists.