Relative Pressure, Relative Equilibrium States, Compensation Functions and Many-to-One Codes Between Subshifts

Abstract

Let $S:X \to X,T:Y \to Y$ be continuous maps of compact metrizable spaces, and let $\pi :X \to Y$ be a continuous surjection with $\pi \circ S = T \circ \pi$. We investigate the notion of relative pressure, which was introduced by Ledrappier and Walters, and study some maximal relative pressure functions that tie in with relative equilibrium states. These ideas are connected with the notion of compensation function, first considered by Boyle and Tuncel, and we show that a compensation function always exists when $S$ and $T$ are subshifts. A function $F \in C(X)$ is a compensation function if $P(S,F + \phi \circ \pi ) = P(T,\phi )\forall \phi \in C(Y)$. When $S$ and $T$ are topologically mixing subshifts of finite type, we relate compensation functions to lifting $T$-invariant measures to $S$-invariant measures, obtaining some results of Boyle and Tuncel. We use compensation functions to describe different types of quotient maps $\pi$. An example is given where no compensation function exists.