On Hutching's conjecture on the topology of the mean action spectrum.

In this talk we present a proof of the following result: for any area-preserving diffeomorphism of the two dimensional closed disk which is neither pseudo-rotation nor periodic, the closure of its mean action spectrum contains an interval with non-zero length. The result can be translated into an analogous result on the frequency of intersections of a Reeb flow with a symplectic surface in a three dimensional contact manifold. This partially answers a question posed by Hutchings asking whether the closure of the mean action spectrum is a connected interval. A possibly stronger expectation would be that the latter set is equal to the closure of the set of asymptotic mean actions of all points in the disk, as it is known for pseudo-rotations. This is a joint work with (Cristofaro-Gardiner)-Pirnapasov-Shelukhin.