Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms

TL;DR

This paper investigates conditions under which surface diffeomorphisms possess orbits with non-zero torsion, demonstrating existence results for area-preserving discs and certain torus diffeomorphisms with rich rotation sets.

Contribution

It establishes the existence of orbits with non-zero torsion for specific classes of surface diffeomorphisms, including area-preserving discs and torus maps with interior rotation sets.

Findings

Existence of orbits with non-zero torsion in area-preserving disc diffeomorphisms

Presence of such orbits in torus diffeomorphisms with non-empty interior rotation set

Results apply to maps isotopic to the identity

Abstract

The present paper concerns the dynamics of surface diffeomorphisms. Given a diffeomorphism $f$ of a surface $S$, the \emph{torsion} of the orbit of a point $z\in S$ is, roughly speaking, the average speed of rotation of the tangent vectors under the action of the derivative of $f$, along the orbit of $z$ under $f$. The purpose of the paper is to identify some situations where there exist measures and orbits with non-zero torsion. We prove that every area preserving diffeomorphism of the disc which coincides with the identity near the boundary has an orbit with non-zero torsion. We also prove that a diffeomorphism of the torus $\mathbb{T}^2$, isotopic to the identity, whose rotation set has non-empty interior, has an orbit with non-zero torsion.