Lyapunov spectra for KMS states on Cuntz-Krieger algebras

Marc Kesseboehmer, Manuel Stadlbauer, Bernd O. Stratmann

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

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 languageEnglish
Pages (from-to)871-893
Number of pages23
JournalMathematische Zeitschrift
Volume256
DOIs
Publication statusPublished - 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

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