Torsion and Linking number for a surface diffeomorphism

TL;DR

This paper establishes a relationship between torsion and linking number for surface diffeomorphisms, providing new proofs and generalizations in the context of twist maps and measure theory.

Contribution

It proves the existence of points with prescribed torsion corresponding to linking numbers and simplifies existing theorems on torsion of measures on the torus.

Findings

Existence of points with torsion equal to given linking numbers.

Simplified proof of Matsumoto and Nakayama's theorem on torsion.

Generalization of linking number estimates for periodic points in twist maps.

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.