Hutchings’ inequality for the Calabi invariant revisited with an application to pseudo-rotations

Hutchings (J Modern Dyn 10:511–539, 2016) used embedded contact homology to show 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. In this article, we extend this to all area-preserving disc diffeomorphisms. Our strategy is to extend the diffeomorphism to a larger disc with precise properties and apply Hutchings’ theorem. As an application, we compute the Calabi invariant of all smooth pseudo-rotations.