Factor map, diamond and density of pressure functions

Abstract

Letting π: X → Y be a one-block factor map and Φ be an almostadditive potential function on X, we prove that if π has diamond, then the pressure P(X, Φ) is strictly larger than P(Y, πΦ). Furthermore, if we define the ratio ρ(Φ) = P(X, Φ)/P(Y, πΦ), then ρ(Φ) > 1 and it can be proved that there exists a family of pairs {(πi,Xi)}ki=1 such that πi: Xi → Y is a factor map between Xi and Y, Xi ⊆ X is a subshift of finite type such that ρ(πi,Φ{pipe}Xi) (the ratio of the pressure function for P(Xi,Φ{pipe}Xi) and P(Y, πΦ)) is dense in [1, ρ(Φ)]. This extends the result of Quas and Trow for the entropy case.