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

We analyze the Block Averaging Transformation applied to the two--dimensional Ising model in the uniqueness region. We discuss the Gibbs property of the renormalized measure and the convergence of renormalized potential under iteration of the map. It turns out that for any temperature $T$ higher than the critical one $T_c$ the renormalized measure is strongly Gibbsian, whereas for $T<T_c$ we have only weak Gibbsianity. Accordingly, we have convergence of the renormalized 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 a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem 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 the presence of a Griffiths' singularity. In this case we use a graded cluster expansion whose minimal scale length is diverging when approaching the coexistence line.