Mathematische Zeitschrift

, Volume 256, Issue 4, pp 871–893 | Cite as

Lyapunov spectra for KMS states on Cuntz-Krieger algebras

  • Marc Kesseböhmer
  • Manuel Stadlbauer
  • Bernd O. Stratmann
Article

Abstract

We study relations between (H,β)-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,β)-KMS states and eigenmeasures of \({\mathcal{L}_{-\beta H}^{\ast}}\) 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.

Keywords

Non-commutative geometry Cuntz-Krieger algebras KMS states Kleinian groups Thermodynamical formalism Fractal geometry Multifractal formalism Lyapunov spectra Markov fibred systems 

Mathematics Subject Classification (2000)

37A55 46L05 46L55 28A80 20H10 

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References

  1. 1.
    Aaronson, J.: An introduction to infinite ergodic theory. Mathematical Surveys and Monographs, vol. 50. American Mathematical Society, Providence (1997)Google Scholar
  2. 2.
    Aaronson J., Denker M. and Urbański M. (1993). Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Am. Math. Soc. 337: 495–548 MATHCrossRefGoogle Scholar
  3. 3.
    Anantharaman-Delaroche C. (1997). Purely infinite C *-algebras arising from dynamical systems. Bull. Soc. Math. France 125: 199–225 MATHMathSciNetGoogle Scholar
  4. 4.
    Anantharaman-Delaroche C., Renault J.: Amenable groupoids. Monographs of L’Enseign. Math. vol. 36, Geneva (2000)Google Scholar
  5. 5.
    Beardon, A.F.: The geometry of discrete groups. Graduate Texts in Mathematics vol. 91, Springer New York (1983)Google Scholar
  6. 6.
    Beardon A.F. and Maskit B. (1974). Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math. 132: 1–12 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics vol. 470. Springer, Berlin (1975)Google Scholar
  8. 8.
    Bowen R. (1980). Hausdorff dimension of quasi-circles. Publ. Math. Inst. Hautes Etud. Sci. 50: 11–25 Google Scholar
  9. 9.
    Bowen R. and Series C. (1979). Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Etud. Sci. 50: 153–170 MATHMathSciNetGoogle Scholar
  10. 10.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics 1. (C *- and W *-algebras, symmetry groups, decomposition of states). Texts and Monographs in Physics, 2nd edn. Springer, New York, (1987)Google Scholar
  11. 11.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics 2. (Equilibrium states. Models in quantum statistical mechanics). Texts and Monographs in physics, 2nd. edn. Springer, Berlin (1987)Google Scholar
  12. 12.
    Cuntz J. and Krieger W. (1980). A class of C *-algebras and topological Markov chains. Invent. Math. 56: 251–268 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Denker, M.: Einführung in die Analysis dynamischer Systeme. Springer-Lehrbuch, Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Mathematics, vol. 527. Springer Heidelberg (1976)Google Scholar
  15. 15.
    Epstein D. and Petronio C. (1994). An exposition of Poincare’s polyhedron theorem. Enseign. Math., II. Ser. 40: 113–170 MATHMathSciNetGoogle Scholar
  16. 16.
    Evans D.E. (1980). On O n. Publ. Res. Inst. Math. Sci., Kyoto Univ. 16: 915–927 MATHGoogle Scholar
  17. 17.
    Exel R. (2004). KMS states for generalized gauge actions on Cuntz-Krieger algebras. Bull. Braz. Math. Soc., New Series 35: 1–12 MATHMathSciNetGoogle Scholar
  18. 18.
    Falconer, K.: Techniques in fractal geometry. Wiley, Chichester (1997)Google Scholar
  19. 19.
    Kac M. (1947). On the notion of recurrence in discrete stochastic processes. Bull. Am. Math. Soc. 53: 1002–1010 MATHCrossRefGoogle Scholar
  20. 20.
    Kerr D. and Pinzari C. (2002). Noncommutative pressure and the variational principle in Cuntz- Krieger-type C *-algebras. J. Funct. Anal. 188: 156–215 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kesseböhmer M. (2001). Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14: 395–409 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kesseböhmer M. and Stratmann B.O. (2004). A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dyn. Sys. 24: 141–170 MATHCrossRefGoogle Scholar
  23. 23.
    Kesseböhmer M. and Stratmann B.O. (2004). Stern-Brocot pressure and multifractal spectra in ergodic theory of numbers. Stochast. Dyn. 4: 77–84 MATHCrossRefGoogle Scholar
  24. 24.
    Kesseböhmer, M., Stratmann, B.O.: A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates. Preprint: arXiv:math.NT/0509603, to appear in J. Reine Angew. Math.Google Scholar
  25. 25.
    Kumjian A. and Renault J. (2006). KMS states on C*-algebras associated to expansive maps. Proc. Am. Math. Soc. 134: 2067–2078 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lott J. (2005). Limit sets as examples in noncommutative geometry. K-theory 54: 283–326 CrossRefMathSciNetGoogle Scholar
  27. 27.
    Matsumoto K., Watani Y. and Yoshida M. (1998). KMS states for gauge actions on C *-algebras associated with subshifts. Math. Z. 228: 489–509 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mauldin, R.D., Urbański, M.: Graph directed Markov systems. Geometry and dynamics of limit sets. Cambridge Tracts in Mathematics, vol. 148. Cambridge University Press, Cambridge (2003)Google Scholar
  29. 29.
    Nicholls, P.J.: The ergodic theory of discrete groups. London Mathematical Society Lecture Notes Series, vol. 143 (1989)Google Scholar
  30. 30.
    Olesen D. and Pedersen G.K. (1978). Some C *-dynamical systems with a single KMS state. Math. Scan. 42: 111–118 MATHMathSciNetGoogle Scholar
  31. 31.
    Patterson S.J. (1976). The limit set of a Fuchsian group. Acta Math. 136: 241–273 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Pesin, Y.B.: Dimension theory in dynamical systems. Contemporary views and applications. Chicago Lectures in Mathematics, University of Chicago Press, Chicago (1997)Google Scholar
  33. 33.
    Renault, J.: AF-equivalence relations and their cocycles. Preprint, arXiv:math.OA/0111182v1Google Scholar
  34. 34.
    Rørdam, M.: Classification of nuclear, simple C * -algebras. In: Classification of nuclear C *-Algebras. Entropy in operator algebras, Encyclopaedia Math. Sci. 126, 1–145 (2002)Google Scholar
  35. 35.
    Ruelle D. (1968). Statistical mechanics of a one-dimensional lattice gas. Comm. Math. Phys. 9: 267–278 MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Stadlbauer M. and Stratmann B.O. (2005). Infinite ergodic theory for Kleinian groups. Ergod. Th. & Dynam. Sys. 25: 1305–1323 MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Sullivan D. (1979). The density at infinity of a discrete group. Institut des Hautes Etudes Scientifiques 50: 171–202 MATHMathSciNetGoogle Scholar
  38. 38.
    Sullivan D. (1984). Entropy, Hausdorff measures old and new and limit sets of geometrically finite Kleinian groups. Acta Math. 153: 259–277 MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Sullivan, D.: Quasiconformal homeomorphisms in dynamics, topology, and geometry. In: Proceedings of the International Congress of Mathematicians, Berkeley, 1986: pp. 1216–1228. Amer. Math. Soc., Providence, (1987)Google Scholar
  40. 40.
    Sullivan D. (1987). Related aspects of positivity in Riemann geometry. J. Diff. Geom. 25: 327–351 MATHMathSciNetGoogle Scholar
  41. 41.
    Yuri M. (1998). Zeta functions for certain non-hyperbolic systems and topological Markov approximations. Ergod. Th. & Dynam. Sys. 18: 1589–1612 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Marc Kesseböhmer
    • 1
  • Manuel Stadlbauer
    • 2
  • Bernd O. Stratmann
    • 3
  1. 1.Fachbereich 3, Mathematik und InformatikUniversität BremenBremenGermany
  2. 2.Institut für Mathematische StochastikGöttingenGermany
  3. 3.Mathematical InstituteUniversity of St. AndrewsSt AndrewsScotland

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