Herman's accumulation conjecture for elliptic equilibria and KAM tori

An elliptic equilibrium is an elliptic fixed point of an analytic Hamiltonian system, and a KAM torus is an invariant quasi-periodic torus of such a system satisfying the KAM conditions. A frequency is Diophantine when it satisfies the relevant Diophantine non-resonance condition. Herman's accumulation conjecture. An elliptic equilibrium with a Diophantine frequency, or a KAM torus, of an analytic Hamiltonian is accumulated by a set of positive measure of KAM tori. The conjecture is known to be true in two degrees of freedom, but remains open in general.