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We analyze the Block Averaging Transformation applied to the two--dimensionalIsing model in the uniqueness region. We discuss the Gibbs property of therenormalized measure and the convergence of renormalized potential underiteration of the map. It turns out that for any temperature $T$ higher than thecritical one $T_c$ the renormalized measure is strongly Gibbsian, whereas for$T<T_c$ we have only weak Gibbsianity. Accordingly, we have convergence of therenormalized potential in a strong sense for $T>T_c$ and in a weak sense for$T<T_c$. Since we are arbitrarily close to the coexistence region we have adiverging characteristic length of the system: the correlation length or thecritical length for metastability, or both. Thus, to perturbatively treat theproblem we use a scale--adapted expansion. The more delicate case is $T<T_c$where we have a situation similar to that of a disordered system in thepresence of a Griffiths' singularity. In this case we use a graded clusterexpansion whose minimal scale length is diverging when approaching thecoexistence line.

Gibbsian properties and convergence of the iterates for the Block Averaging Transformation