We consider an infinite-to-one factor map from an irreducible shift of finitetype X to a sofic shift Y. A compensation function relates equilibrium stateson X to equilibrium states on Y. The p-Dini condition is given as a way ofmeasuring the smoothness of a continuous function, with 1-Dini corresponding tofunctions with summable variation. Two types of compensation functions aredefined in terms of this condition. We show that the relative equilibriumstates of a 1-Dini function f over a fully supported invariant measure on Y arethemselves fully supported, and have positive relative entropy. We then showthat there exists a compensation function which is p-Dini for all p > 1 whichhas relative equilibrium states supported on a finite-to-one subfactor.
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