Raphael Krikorian: On local integrability of real analytic conservative diffeomorphisms in dimension 2

Abstract:

Let TWIP be the class of smooth diffeomorphisms of the 2-disk or the 2-cylinder that admit a non resonant/Diophantine elliptic equilibrium, have the intersection property and satisfy a twist condition. Birkhoff Normal Form and KAM theory tell us that this elliptic equilibrium is accumulated by a positive measure set of invariant KAM circles. I shall present in this talk a proof of the following result: any real analytic TWIP diffeomorphism is accumulated in the real analytic topology by real analytic TWIP’s that are locally integrable at the origin, which means that an neighborhood of the origin (depending on the order of approximation) is fully covered by KAM invariant circles. The proof is based on KAM theory and arguments from sheaf theory and deformation of complex structures.