Relative pressure functions and their equilibrium states

Abstract For a subshift $(X, \sigma _{X})$ and subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on X , we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim _{n\rightarrow \infty }(1/{n})\int \log f_{n}\, d\kern-1pt\mu =\int h \,d\kern-1pt\mu $ every invariant measure $\mu . this purpose, first necessary sufficient F}$ to be an asymptotically additive in terms certain properties periodic points. factor map $\pi : X\rightarrow Y$ where is irreducible shift finite type $(Y, _{Y})$ subshift, applying our results obtained by Cuneo [Additive, almost potential sequences are equivalent. Comm. Math. Phys. 37 (3) (2020), 2579–2595] sequences, with regard associated relative pressure function. This leads characterization continuous compensation function between subshifts. As application, projection \mu weak Gibbs type.