Radon–nikodym Representations of Cuntz–krieger Algebras and Lyapunov Spectra for Kms States

We study relations between (H,β)–KMS states on Cuntz–Krieger algebras and the dual of the Perron–Frobenius operator L∗ −βH . 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 L∗ −βH for the eigenvalue 1. We then consider representations of Cuntz–Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are ∗–isomorphic to the given Cuntz–Krieger algebra. Finally, we 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.