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

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

2006-01-15

arxiv.org/abs/math/0601354v2

Mathematische Zeitschrift 256 (2007), no. 4, 871-893

Mathematics

Dynamical Systems

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

Scientific paper

10.1007/s00209-007-0110-y

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 $G$.

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