Title: A general renormalization procedure on the one-dimensional lattice and decay of correlations,
Abstract: We present a general form of Renormalization operator $\mathcal{R}$ acting on
potentials $V:\{0,1\}^\mathbb{N} \to \mathbb{R}$. We exhibit the analytical
expression of the fixed point potential $V$ for such operator $\mathcal{R}$.
This potential can be expressed in a naturally way in terms of a certain
integral over the Hausdorff probability on a Cantor type set on the interval
$[0,1]$. This result generalizes a previous one by A. Baraviera, R. Leplaideur
and A. Lopes where the fixed point potential $V$ was of Hofbauer type.
For the potentials of Hofbauer type (a well known case of phase transition)
the decay is like $n^{-\gamma}$, $\gamma>0$.
Among other things we present the estimation of the decay of correlation of
the equilibrium probability associated to the fixed potential $V$ of our
general renormalization procedure. In some cases we get polynomial decay like
$n^{-\gamma}$, $\gamma>0$, and in others a decay faster than $n \,e^{ -\,
\sqrt{n}}$, when $n \to \infty$.
The potentials $g$ we consider here are elements of the so called family of
Walters potentials on $\{0,1\}^\mathbb{N} $ which generalizes the potentials
considered initially by F. Hofbauer. For these potentials some explicit
expressions for the eigenfunctions are known.
In a final section we also show that given any choice $d_n \to 0$ of real
numbers varying with $n \in \mathbb{N}$ there exist a potential $g$ on the
class defined by Walters which has a invariant probability with such numbers as
the coefficients of correlation (for a certain explicit observable function).