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In an earlier work we have considered a lattice system, modelled by a directed and weighted graph G (whose vertices are the spins and its adjacency matrix M will be given by the system transition rules), One of the characteristics of the system is a matrix A(q), with A(0) = M, whose values depend on the system interactions. To A(q) is associated the system free energy FA(q), defined as the spectral radius of A(q), which is related with the energy usually introduced in Statistical Mechanics as proportional to the logarithm of the partition function. In that mentioned article we considered Gibbs ensembles supported on periodic configurations which are equilibrium states provided conditions essentially imposed on the matrix. Herein we analyze the construction of ensembles supported on periodic configurations for systems semi-conjugated to a given one which has equilibrium states.