A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations

TL;DR

This paper provides a finite-dimensional proof that the Calabi invariant equals the rotation number for certain pseudo-rotations of the disk, using generating functions and gradient flow dynamics, extending Hutchings' earlier work.

Contribution

It offers a new finite-dimensional proof of Hutchings' result, avoiding Embedded Contact Homology techniques and employing generating functions and gradient flow analysis.

Findings

Calabi invariant equals rotation number for $C^1$ pseudo-rotations

New proof technique using generating functions and gradient flows

Extends Hutchings' result with a different mathematical approach

Abstract

We prove that the Calabi invariant of a $C^1$ pseudo-rotation of the unit disk, that coincides with a rotation on the unit circle, is equal to its rotation number. This result has been shown some years ago by Michael Hutchings (under very slightly stronger hypothesis). While the original proof used Embedded Contact Homology techniques, the proof of this article uses generating functions and the dynamics of the induced gradient flow.