  We consider a family of potentials f, derived from the Hofbauer potentials,
on the symbolic space Omega=\{0,1\}^\mathbb{N} and the shift mapping $\sigma$
acting on it. A Ruelle operator framework is employed to show there is a phase
transition when the temperature varies in the following senses: the pressure is
not analytic, there are multiple eigenprobabilities for the dual of the Ruelle
operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic
Limit is not unique. Additionally, we explicitly calculate the critical points
for these phase transitions. Some examples which are not of Hofbauer type are
also considered. The non-uniqueness of the Thermodynamic Limit is proved by
considering a version of a Renewal Equation. We also show that the correlations
decay polynomially and that each one of these Hofbauer potentials is a fixed
point for a certain renormalization transformation.
