Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982). Theorem A (Main result). For every analytic and closed, oriented surface (S, Leb), there exists a symplectic analytic map f ∈ Diff^ω_Leb (S) of any isotopy class, such that