Mean action and the Calabi invariant

TL;DR

This paper establishes a relationship between the Calabi invariant and the mean action of periodic orbits for area-preserving disk diffeomorphisms, introducing a new contact homology filtration.

Contribution

It introduces a novel filtration on embedded contact homology based on a transverse knot, applicable to various contact homology theories.

Findings

If Calabi invariant < boundary rotation number, then infimum of mean action over periodic orbits ≤ Calabi invariant.

The new filtration on embedded contact homology can be applied to other contact homology theories.

The proof provides a link between dynamical properties and contact homology invariants.

Abstract

Given an area-preserving diffeomorphism of the closed unit disk which is a rotation near the boundary, one can naturally define an "action" function on the disk which agrees with the rotation number on the boundary. The Calabi invariant of the diffeomorphism is the average of the action function over the disk. Given a periodic orbit of the diffeomorphism, its "mean action" is defined to be the average of the action function over the orbit. We show that if the Calabi invariant is less than the boundary rotation number, then the infimum over periodic orbits of the mean action is less than or equal to the Calabi invariant. The proof uses a new filtration on embedded contact homology determined by a transverse knot, which might be of independent interest. (An analogue of this filtration can be defined for any other version of contact homology in three dimensions that counts holomorphic curves.)