Factor Map, Diamond and Density of Pressure Functions

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|Xi ) (the ratio of the pressure function for P (Xi,Φ|Xi ) and P (Y, πΦ)) is dense in [1, ρ(Φ)]. This extends the result of Quas and Trow for the entropy case.