  For the subshift of finite type $\S=\{0,1,2\}^{\N}$ we study the convergence
at temperature zero of the Gibbs measure associated to a non-locally constant
H\"older potential which admits only two maximizing measures. These measures
are Dirac measures at two different fixed points. The potential is flattest at
one of these two fixed points. The question we are interested is: which of
these probabilities the invariant Gibbs state will select when temperature goes
to zero? We prove that on the one hand the Gibbs measure converges, and at the
other hand it does not necessarily converge to the measures where the potential
is the flattest. We consider a family of potentials of the above form; for some
of them there is the selection of a convex combination of the two Dirac
measures, and for others there is a selection of the Dirac measure associated
to the flattest point. In the first case this is contrary to what was expected
if we consider the analogous problem in Aubry-Mather theory by N. Anantharaman,
R. Iturriaga, P. Padilla and H. Sanchez-Morgado. The invariant probability is
obtained by the junction of the eigen-function and the eigen-probability. A
curious phenomena that happens in our examples is that the eigen-measure and
the eigen-function have opposite behavior. When $\beta \to \infty$, the
eigen-measure became exponential bigger around $0^\infty$, when compared to
points around $1^\infty$. For the eigen-function the opposite happens.
