An SMB approach for pressure representation in amenable virtually orderable groups

TL;DR

This paper develops a new pressure representation method for certain group actions on subshifts, extending previous results to more general groups and relaxed conditions using an SMB theorem.

Contribution

It introduces a pressure representation approach for amenable virtually orderable groups, generalizing prior work to broader settings with less restrictive assumptions.

Findings

Provides necessary and sufficient conditions for pressure representation.

Extends SMB theorem applications to new group settings.

Generalizes prior results to more flexible assumptions.

Abstract

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.