Torsion and Linking number for a surface diffeomorphism

Abstract

For a $\mathcal{C}^1$ diffeomorphism $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ 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.