Mean action of periodic orbits of area-preserving annulus diffeomorphisms

TL;DR

This paper investigates the relationship between the Calabi invariant and the mean action of periodic orbits in area-preserving annulus diffeomorphisms, establishing bounds under certain boundary conditions.

Contribution

It proves that for specific boundary conditions, the infimum of the mean action of periodic orbits is bounded above by the Calabi invariant.

Findings

The infimum of mean actions is less than or equal to the Calabi invariant under given conditions.

The result applies when the diffeomorphism is a rotation near the boundary.

The paper extends understanding of the action spectrum in area-preserving annulus maps.

Abstract

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary of the annulus, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.