  We propose a new model of ergodic optimization for expansive dynamical
systems: the holonomic setting. In fact, we introduce an extension of the
standard model used in this theory. The formulation we consider here is quite
natural if one wants a meaning for possible variations of a real trajectory
under the forward shift. In another contexts (for twist maps, for instance),
this property appears in a crucial way.
  A version of the Aubry-Mather theory for symbolic dynamics is introduced. We
are mainly interested here in problems related to the properties of maximizing
probabilities for the two-sided shift. Under the transitive hypothesis, we show
the existence of sub-actions for Holder potentials also in the holonomic
setting. We analyze then connections between calibrated sub-actions and the
Mane potential. A representation formula for calibrated sub-actions is
presented, which drives us naturally to a classification theorem for these
sub-actions. We also investigate properties of the support of maximizing
probabilities.
