Lyapunov spectra for KMS states on Cuntz-Krieger algebras

We study relations between (H,β)-KMS states on Cuntz-Krieger algebras and the dual of the Perron-Frobenius operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}_{-\beta H}^{\ast}}$$\end{document} . Generalising the well-studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one-one correspondence between (H,β)-KMS states and eigenmeasures of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}_{-\beta H}^{\ast}}$$\end{document} for the eigenvalue 1. We then apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups G which may have parabolic elements. We show that for the Cuntz-Krieger algebra arising from G there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen-Series map associated with G. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of G. If G has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with G.