Skip to main content
SpringerLink
Log in

Thermodynamics, dimension and the Weil–Petersson metric

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Abenda, S., Moussa, P., Osbaldestin, A.H.: Multifractal dimensions and thermodynamical description of nearly-circular Julia sets. Nonlinearity 12, 19–40 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bers, L.: Uniformization and moduli (1960). In: Selected Works of Lipman Bers, Part 1, pp. 299–308. Am. Math. Soc. (1998)

  3. Bonahon, F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92, 139–162 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bowen, R., Series, C.: Markov maps associated with Fuchsian groups. Publ. Math., Inst. Hautes Étud. Sci. 50, 153–170 (1978)

    Google Scholar 

  5. Bridgeman, M., Taylor, E.C.: Patterson–Sullivan measures and quasi-conformal deformations. Commun. Anal. Geom. 13, 561–589 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Bridgeman, M., Taylor, E.C.: An extension of the Weil–Petersson metric to the quasi-Fuchsian space. Math. Ann. (to appear)

  7. Burger, M.: Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not. 7, 217–225 (1993)

    Article  MathSciNet  Google Scholar 

  8. DeMarco, L.: Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326, 43–73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Denker, M., Philipp, W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dyn. Syst. 4, 541–552 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eskin, A., McMullen, C.: Mixing, counting and equidistribution in Lie groups. Duke Math. J. 71, 181–209 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fornaess, J.E., Sibony, N.: Harmonic currents of finite energy and laminations. Geom. Funct. Anal. 15, 962–1003 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gardiner, F.P., Sullivan, D.P.: Symmetric structures on a closed curve. Am. J. Math. 114, 683–736 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gardiner, F.P., Sullivan, D.P.: Lacunary series as quadratic differentials in conformal dynamics. In: The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions. Contemp. Math., vol. 169, pp. 307–330. Am. Math. Soc. (1994)

  14. Garnett, L.: Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51, 285–311 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Knapp, A.W.: Representation Theory of Semisimple Groups. Princeton University Press (1986)

  16. Lyubich, M., Minsky, Y.: Laminations in holomorphic dynamics. J. Differ. Geom. 47, 17–94 (1997)

    MathSciNet  MATH  Google Scholar 

  17. Makarov, N.G.: On the distortion of boundary sets under conformal mappings. Proc. Lond. Math. Soc., III. Ser. 51, 369–384 (1985)

    Article  MATH  Google Scholar 

  18. Mañé, R.: The Hausdorff dimension of invariant probabilities of rational maps. In: Dynamical Systems, Valparaiso 1986. Lect. Notes Math., vol. 1331, pp. 86–117. Springer (1988)

  19. Manning, A.: The dimension of the maximal measure for a polynomial map. Ann. Math. 119, 425–430 (1984)

    Article  MathSciNet  Google Scholar 

  20. Martin, N.F.G.: On finite Blaschke products whose restrictions to the unit circle are exact endomorphisms. Bull. Lond. Math. Soc. 15, 343–348 (1983)

    Article  MATH  Google Scholar 

  21. McMullen, C.: Automorphisms of rational maps. In: Holomorphic Functions and Moduli I, pp. 31–60. Springer (1988)

  22. McMullen, C.: Complex Dynamics and Renormalization. Ann. Math. Stud., vol. 135. Princeton University Press (1994)

  23. McMullen, C.: Rational maps and Teichmüller space. In: Havin, V.P., Nikolskii, N. K. (eds.) Linear and Complex Analysis Problem Book. Lect. Notes Math., vol. 1574, pp. 430–433. Springer (1994)

  24. McMullen, C.: The classification of conformal dynamical systems. In: Current Developments in Mathematics, 1995, pp. 323–360. International Press (1995)

  25. McMullen, C.: Renormalization and 3-Manifolds which Fiber over the Circle. Ann. Math. Stud., vol. 142. Princeton University Press (1996)

  26. McMullen, C.: Complex earthquakes and Teichmüller theory. J. Am. Math. Soc. 11, 283–320 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. McMullen, C.: Hausdorff dimension and conformal dynamics III: Computation of dimension. Am. J. Math. 120, 691–721 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Melbourne, I. Török, A.: Statistical limit theorems for suspension flows. Isr. J. Math. 144, 191–209 (2004)

    Article  MATH  Google Scholar 

  29. de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer (1993)

  30. Milnor, J.: Pasting together Julia sets: a worked out example of mating. Exp. Math. 13, 55–92 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Astérisque, vol. 187–188 (1990)

  32. Pollicott, M.: On the mixing of Axiom A attracting flows and a conjecture of Ruelle. Ergodic Theory Dyn. Syst. 19, 535–548 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pommerenke, C.: Boundary Behavior of Conformal Maps. Springer (1992)

  34. Ruelle, D.: Thermodynamic Formalism: the Mathematical Structures of Classical Equilibrium Statistical Mechanics. Addison-Wesley (1978)

  35. Ruelle, D.: Repellers for real analytic maps. Ergodic Theory Dyn. Syst. 2, 99–107 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ruelle, D.: Flots qui ne mélangent pas exponentiellement. C. R. Acad. Sci., Paris, Sér. I, Math. 296, 191–193 (1983)

    MathSciNet  MATH  Google Scholar 

  37. Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Proc. Sympos. Pure Math., Vol. VIII, pp. 1–15. Am. Math. Soc. (1965)

  38. Sinai, Y.G.: The central limit theorem for geodesic flows on manifolds of constant negative curvature. Sov. Math. Dokl. 1, 983–987 (1960)

    MathSciNet  MATH  Google Scholar 

  39. Sullivan, D.: Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups. Acta Math. 155, 243–260 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sullivan, D.: Quasiconformal homeomorphisms in dynamics, topology and geometry. In: Proceedings of the International Conference of Mathematicians, pp. 1216–1228. Am. Math. Soc. (1986)

  41. Sullivan, D.: Bounds, quadratic differentials and renormalization conjectures. In: Browder, F. (ed.) Mathematics into the Twenty-first Century: 1988 Centennial Symposium, August 8–12, pp. 417–466. Am. Math. Soc. (1992)

  42. Sullivan, D.: Linking the universalities of Milnor–Thurston, Feigenbaum and Ahlfors–Bers. In: Goldberg, L.R., Phillips, A.V. (eds.) Topological Methods in Modern Mathematics, pp. 543–564. Publish or Perish, Inc. (1993)

  43. Tan, L.: Matings of quadratic polynomials. Ergodic Theory Dyn. Syst. 12, 589–620 (1992)

    MATH  Google Scholar 

  44. Weil, A.: [1958b] On the moduli of Riemann surfaces. In: Oeuvres Scient., vol. II, pp. 381–389. Springer (1980)

  45. Wolpert, S.: An asymptotic formula for the Taylor coefficients of automorphic forms. Proc. Am. Math. Soc. 78, 485–491 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wolpert, S.: Thurston’s Riemannian metric for Teichmüller space. J. Differ. Geom. 23, 143–174 (1986)

    MathSciNet  MATH  Google Scholar 

  47. Zinsmeister, M.: Formalisme thermodynamique et systèmes dynamiques holomorphes. Société Mathématique de France (1996)

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Harvard University, 1 Oxford St, Cambridge, MA, 02138-2901, USA

    Curtis T. McMullen

Authors
  1. Curtis T. McMullen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Curtis T. McMullen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McMullen, C. Thermodynamics, dimension and the Weil–Petersson metric . Invent. math. 173, 365–425 (2008). https://doi.org/10.1007/s00222-008-0121-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-008-0121-2

Keywords

Search

Navigation