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

TL;DR

This paper explores the relationship between KMS states on Cuntz--Krieger algebras, eigenmeasures of the Perron--Frobenius operator, and Lyapunov spectra, applying these concepts to Kleinian groups and their limit sets.

Contribution

It establishes a correspondence between KMS states and eigenmeasures for expansive systems, and applies this to multifractal analysis of Kleinian group limit sets.

Findings

One-to-one correspondence between KMS states and eigenmeasures for expansive systems.

Representation of Cuntz--Krieger algebras induced by Markov fibred systems is isomorphic under irreducibility.

Derived formulas for Hausdorff dimensions related to KMS states and Lyapunov spectra.

Abstract

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