Maria Saprykina: Isolated elliptic fixed points for smooth Hamiltonians

KAM theory asserts that generically an elliptic fixed point of a
Hamiltonian system is stable in a probabilistic sense, or KAM stable,
which means that the fixed point is accumulated by a positive measure
set of invariant Lagrangian tori.

It was conjectured by M. Herman in his ICM98 lecture that for analytic
Hamiltonians, KAM stability holds in a neighborhood of an elliptic
fixed point if its frequency vector is assumed to be Diophantine. The
conjecture is known to be true in two degrees of freedom, but remains
open in general. Partial results in this direction were recently
obtained Eliasson, Fayad and Krikorian.

Below analytic regularity, Herman proved that KAM stability of a
Diophantine equilibrium holds without any twist condition for
infinitely smooth systems in 2 degrees of freedom.  In his ICM98
lecture, Herman announced that KAM stability of Diophantine equilibria
does not hold for smooth Hamiltonians in 4 or more degrees of freedom,
without giving any clew about the possible counter-examples. He also
wrote that nothing was known about KAM stability of Diophantine
equilibria for smooth Hamiltonians in 3 degrees of freedom.

In this talk, which is based on a joint work with Bassam Fayad, we
settle this problem by presenting examples of smooth Hamiltonians in 3
or more degrees of freedom, having non KAM stable elliptic equilibria
with arbitrary frequency.