Relative entropy functions for factor maps between subshifts

Let $(X, S)$ and $(Y, T)$ be topological dynamical systems and $\pi : X \rightarrow Y$ a factor map. A function $F \in C (X)$ is a compensation function for $\pi$ if $P (F + \phi \circ \pi ) = P (\phi)$ for all $\phi \in C(Y)$. For a factor code between subshifts of finite type, we analyze the associated relative entropy function and give a necessary condition for the existence of saturated compensation functions. Necessary and sufficient conditions for a map to be a saturated compensation function will be provided.