Hutchings’ inequality for the Calabi invariant

Hutchings proved the following for area-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: if the asymptotic mean action on the boundary is greater than the Calabi invariant, then the infimum of the mean action of the periodic points is less than or equal to the Calabi invariant. I will explain how to extend this result to any symplectic disc diffeomorphisms and also introduce a more general result for area-preserving disc diffeomorphisms with only one periodic point. This is joint work with David Bechara, Barney Bramham, and Patrice Le Calvez.