This work is devoted to the study of processes generated by random substitutions over a finite alphabet. We prove, under mild conditions on the substitution's rule, the existence of a unique process which remains invariant under the substitution, and exhibiting polynomial decay of correlations. For constantlength substitutions, we go further by proving that the invariant state is precisely a Gibbs measure which can be obtained as the projective limit of its natural Markovian approximations. We close the paper with a class of substitutions whose invariant state is the unique Gibbs measure for a hierarchical two-body interaction.

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to thed-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of g-measures, it allows to assess the continuity of the entropy at g-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite g-functions, to the preservation at the limit, of certain ergodic properties for the associate g-measures.

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