Herman's analytic positive entropy conjecture

Let the disk carry its standard analytic symplectic structure, and let an analytic symplectic perturbation of the identity mean an analytic symplectic diffeomorphism sufficiently close to the identity. Positive metric entropy means that the map has positive metric entropy with respect to the symplectic area measure. Herman's analytic positive entropy conjecture. There exists an analytic and symplectic perturbation of the identity of the disk with positive metric entropy. Herman's original smooth positive entropy conjecture was proved in the cited work of Buzzi and Turaev; the analytic counterpart is presented here as the remaining conjectural strengthening.