On the Gibbs measures of commuting one-sided subshifts of finite type (Q1568367)

The authors generalize a result on Parry measures to prove a result on the equivalence of Gibbs measures on mixing one-sided subshifts of finite type. For such a space \((X,{\mathcal S})\), if \(f\) is a real-valued function, we can define the coboundary operator \(\tau s(f)= f-f\circ{\mathcal S}\). The main result derived by the authors is the following: Theorem: Let \({\mathcal S}, {\mathcal T}:X\to X\) be commuting maps such that \((X,{\mathcal S})\) and \((X,{\mathcal T})\) are mixing one-sided subshifts of finite type, and assume that \(\phi, \psi:X\to\mathbb{R}\) are continuous functions with summable variation. Then the Gibbs measures \(\mu{\mathcal S}_\varphi\) and \(\mu{\mathcal T}_\psi\) are equal if and only if there exists a continuous function \(\omega\) on \(X\) such that \(\tau S\varphi -\tau T\psi= \tau S \tau T\omega\). Moreover, if \(\varphi\), \(\psi\) are Hölder continuous, then so is \(\omega\).