An SMB approach for pressure representation in amenable virtually orderable groups

Given a countable discrete amenable virtually orderable group $G$ acting by translations on a $G$-subshift $X \subseteq S^G$ and an absolutely summable potential $\Phi$, we present a set of conditions to obtain a special integral representation of pressure $P(\Phi)$. The approach is based on a Shannon-McMillan-Breiman (SMB) type theorem for Gibbs measures due to Gurevich-Tempelman (2007), and generalizes results from Gamarnik-Katz (2009), Helvik-Lindgren (2014), and Marcus-Pavlov (2015) by extending the setting to other groups besides $\mathbb{Z}^d$, by relaxing the assumptions on $X$ and $\Phi$, and by using sufficient convergence conditions in a mean --instead of a uniform-- sense. Under the fairly general context proposed here, these same conditions turn out to be also necessary.