  We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras
and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$.
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 $(H,\beta)$--KMS states and eigenmeasures of
$\mathcal{L}_{-\beta H}^{*}$ 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 $\ast$--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 $G$.

Dynamical Systems;Operator Algebras