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

An SMB approach for pressure representation in amenable virtually orderable groups