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We show that an arbitrary factor map $\pi:X \to Y$ on an irreducible subshiftof finite type is a composition of a finite-to-one factor code and a classdegree one factor code. Using this structure theorem on infinite-to-one factorcodes, we then prove that any equilibrium state $\nu$ on $Y$ for a potentialfunction of sufficient regularity lifts to a unique measure of maximal relativeentropy on $X$. This answers a question raised by Boyle and Petersen (for liftsof Markov measures) and generalizes the earlier known special case offinite-to-one factor codes.

Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states