  In this paper, we show a new relation between phase transition in
one-dimensional Statistical Mechanics and the multiplicity of the dimension of
the space of harmonic functions for an extension of the classical transfer
operator. We accomplish this by extending the classical Ruelle-Perron-Frobenius
theory to the realm of low regular potentials. This is done by establishing
finer properties of the associated conformal measures and thoroughly developing
a method to obtain information on the maximal eigenspace of a suitably
constructed family of Markov Processes. Our results are valid in the setting of
finite and infinite alphabets. Several new applications are given to illustrate
the theory. For example, we determine the support of a large class of
equilibrium states associated with low regular potentials, including ones
allowing phase transition. Additionally, we prove a version of the Functional
Central Limit Theorem for equilibrium states. A remarkable aspect of this
result is that it does not require the spectral gap property of the associated
transfer operator. It is valid for long-range spins systems that might not be
positively correlated and for non-local observables.
