The Calabi invariant for Hamiltonian diffeomorphisms of the unit disk

TL;DR

This paper extends the Calabi invariant to a broader class of disk diffeomorphisms and computes it for specific cases, enhancing understanding of symplectic invariants in disk dynamics.

Contribution

The paper introduces an extension of the Calabi invariant to $C^1$ diffeomorphisms of the closed disk preserving the symplectic form, and computes it for certain rigid diffeomorphisms.

Findings

Calabi invariant extended to $C^1$ diffeomorphisms of the closed disk

Explicit calculations of the Calabi invariant for specific diffeomorphisms

Insights into rigidity properties of disk diffeomorphisms

Abstract

In this article, we study the Calabi invariant on the unit disk usually defined on compactly supported Hamiltonian diffeomorphisms of the open disk. In particular we extend the Calabi invariant to the group of $C^1$ diffeomorphisms of the closed disk which preserves the standard symplectic form. We also compute the Calabi invariant for some diffeomorphisms of the disk which satisfies some rigidity hypothesis.