Factor map, diamond and density of pressure functions

Abstract:

Letting $\pi :X\rightarrow Y$ be a one-block factor map and $\Phi$ be an almost-additive potential function on $X,$ we prove that if $\pi$ has diamond, then the pressure $P(X,\Phi )$ is strictly larger than $P(Y,\pi \Phi )$. Furthermore, if we define the ratio $\rho (\Phi )=P(X,\Phi )/P(Y,\pi \Phi )$, then $\rho (\Phi )>1$ and it can be proved that there exists a family of pairs $\left \{ (\pi _{i},X_{i})\right \} _{i=1}^{k}$ such that $\pi _{i}:X_{i} \rightarrow Y$ is a factor map between $X_{i}$ and $Y$, $X_{i}\subseteq X$ is a subshift of finite type such that $\rho (\pi _{i},\Phi |_{X_{i}})$ (the ratio of the pressure function for $P(X_{i},\Phi |_{X_{i}})$ and $P(Y,\pi \Phi )$) is dense in $[1,\rho (\Phi )]$. This extends the result of Quas and Trow for the entropy case.