Torsion and linking number for a surface diffeomorphism

Abstract

For a C 1 diffeomorphism f : R 2 → R 2 isotopic to the identity, we prove that for any value l ∈ 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 ∈ R. As an outcome, we give a much simplier proof of a theorem by Matsumoto and Nakayama concerning torsion of measures on 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.

On the set of points of zero torsion for negative-torsion maps of the annulus

“…Corollary 3.1 (Corollary 3.1 in [Flo19b]). Let F : R 2 → R 2 be a C 1 diffeomorphism isotopic to the identity and let I be an isotopy joining the identity to F 1 = F .…”

“…The torsion at finite time does not depend on the choice of the continuous determination of the oriented angle function. Moreover, it is independent from the choice of the isotopy joining the identity to f , see Proposition 2.5 in [Flo19b]. The (asymptotic) torsion, whenever it exists, does not depend on the tangent vector used to calculate the finite time torsion.…”

“…Example 2.2. Every positive twist map is a negative-torsion map (see [Flo19b]). Actually the notion of negative-torsion map is equivalent to the notion of positive tilt map: see [Hu98], [GR13] and [Flo19a].…”

“…We recall here some properties of finite-time torsion. We refer to [BB13], [Flo19b] and [Flo19a] for the proofs.…”

“…Concerning the relation between torsion and linking number, we recall here Corollary 3.1 in [Flo19b]. The torsion is calculated with respect to the standard trivialization.…”