Rafael de la Llave, Enrico Valdinoci
Critical points
inside the gaps of ground state laminations
in statistical mechanics
(372K, PostScript)

ABSTRACT.  We consider models of interacting particles
situated in the points of a discrete set.
The state of each particle is determined by a real
variable.
The particles are interacting with each other
and  we are interested in ordered ground states.
Under the assumption that the set
and the interaction are symmetric under
the action of a group -- which satisfies some
mild assumption --, that the interaction is ferromagnetic,
as well as periodic under addition of integers,
it was shown in a previous paper that there are many ground
states that satisfy an order property called selfconforming
or Birkhoff.
Under the assumption that the interaction is finite range, we
show that either the ground states form a one dimensional family
or that there are other Birkhoff critical points which are
not ground states.
In the case that the set is just a one dimensional lattice and that the 
interaction is just nearest neighbor, this
generalizes Mather's criterion for the existence of invariant circles.