Instability of Renormalization-Group Pathologies under decimation

We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non Gibbsian measure, Gibbsiannes may be restored by applying an extra decimation trasformation. This fact is illustrated in details for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.