## Abstract

We study relations between (H,beta)-KMS states on Cuntz-Krieger algebras and the dual of the Perron-Frobenius operator 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 L--beta H(*) for the eigenvalue 1. We then 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.

Original language | English |
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Pages (from-to) | 871-893 |

Number of pages | 23 |

Journal | Mathematische Zeitschrift |

Volume | 256 |

DOIs | |

Publication status | Published - Aug 2007 |

## Keywords

- non-commutative geometry
- Cuntz-Krieger algebras
- KMS states
- Kleinian groups
- thermodynamical formalism
- fractal geometry
- multifractal formalism
- Lyapunov spectra
- Markov fibred systems
- C-ASTERISK-ALGEBRAS
- FINITE KLEINIAN-GROUPS
- DYNAMICAL-SYSTEMS
- ERGODIC-THEORY
- GAUGE ACTIONS
- MAPS