# Torsion and linking number for a surface diffeomorphism

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## 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 simplier proof of a theorem by Matsumoto and Nakayama concerning torsion of measures 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.

## Notes

### Acknowledgements

The author is extremely grateful to Professor Marie-Claude Arnaud and Andrea Venturelli for their precious advices to improve the text and for many stimulating discussions. The author acknowledges the anonymous referee for his or her useful remarks and observations.

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