KMS states for generalized gauge actions on C*-algebras associated with self-similar sets

Abstract

Given a self-similar $K$ set defined from an iterated function system $Γ=(γ_1,\ldots,γ_n)$ and a set of function $H=\{h_i:K\to\mathbb{R}\}_{i=1}^d$ satisfying suitable conditions, we define a generalized gauge action on Kawjiwara-Watatani algebras $\mathcal{O}_Γ$ and their Toeplitz extensions $\mathcal{T}_Γ$. We then characterize the KMS states for this action. For each $β\in(0,\infty)$, there is a Ruelle operator $\mathcal{L}_{H,β}$ and the existence of KMS states at inverse temperature $β$ is related to this operator. The critical inverse temperature $β_c$ is such that $\mathcal{L}_{H,β_c}$ has spectral radius 1. If $β<β_c$, there are no KMS states on $\mathcal{O}_Γ$ and $\mathcal{T}_Γ$; if $β=β_c$, there is a unique KMS state on $\mathcal{O}_Γ$ and $\mathcal{T}_Γ$ which is given by the eigenmeasure of $\mathcal{L}_{H,β_c}$; and if $β>β_c$, including $β=\infty$, the extreme points of the set of KMS states on $\mathcal{T}_Γ$ are parametrized by the elements of $K$ and on $\mathcal{O}_Γ$ by the set of branched points.