Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

Abstract

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

30 pages, minor changes in the proofs of Theorem 3.9 and Fact 9

References in corpus (2)

• A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates
• AF-equivalence relations and their cocycles

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• Multiresolution analysis for Markov Interval Maps