  We generalize several results of the classical theory of Thermodynamic
Formalism by considering a compact metric space $M$ as the state space. We
analyze the shift acting on $M^\mathbb{N}$ and consider a general a-priori
probability for defining the Transfer (Ruelle) operator. We study potentials
$A$ which can depend on the infinite set of coordinates in $M^\mathbb{N}.$
  We define entropy and by its very nature it is always a nonpositive number.
The concepts of entropy and transfer operator are linked. If M is not a finite
set there exist Gibbs states with arbitrary negative value of entropy.
Invariant probabilities with support in a fixed point will have entropy equal
to minus infinity. In the case $M=S^1$, and the a-priori measure is Lebesgue
$dx$, the infinite product of $dx$ on $(S^1)^\mathbb{N}$ will have zero
entropy.
  We analyze the Pressure problem for a H\"older potential $A$ and its relation
with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other
things we analyze the case where temperature goes to zero and we show some
selection results. Our general setting can be adapted in order to analyze the
Thermodynamic Formalism for the Bernoulli space with countable infinite
symbols. Moreover, the so called XY model also fits under our setting. In this
last case M is the unitary circle $S^1$. We explore the differentiable
structure of $(S^1)^\mathbb{N}$ by considering potentials which are of class
$C^1$ and we show some properties of the corresponding main eigenfunctions.
