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