We study compensation functions for an infinite-to-one factor code $\pi : X \to Y$ where $X$ is a shift of finite type. 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. Given a fully-supported invariant measure $\nu$ on $Y$, we show that the relative equilibrium states of a $1$-Dini function $f$ over $\nu$ 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. / Graduate / 0405 / antoniol@uvic.ca