Skip to main content
SpringerLink
Log in
Book cover

Contributions in Analytic and Algebraic Number Theory pp 171–195Cite as

The Patterson Measure: Classics, Variations and Applications

  • Conference paper
  • First Online:

  • 1160 Accesses

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 9))

Abstract

This survey is dedicated to S. J. Patterson’s 60th birthday in recognition of his seminal contribution to measurable conformal dynamics and fractal geometry. It focuses on construction principles for conformal measures for Kleinian groups, symbolic dynamics, rational functions and more general dynamical systems, due to Patterson, Bowen-Ruelle, Sullivan and Denker-Urbański.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    R is said to be nonsingular with respect to μ1and μ2, if for each measurable set E ⊂ Ω 2one has that \({\mu }_{1}({R}^{-1}(E)) = 0\) if and only if μ2(E) = 0.

  2. 2.

    The ω limit sets of critical points in the Fatou set are attracting or parabolic cycles and the ω limit set of critical points c in the Julia set are compact in \(\mathbb{C} \setminus \{ c\}\) (resp. T n (c) = ∞, for some n ≥ 1).

  3. 3.

    That is of the form T(z) = λexp (z).

  4. 4.

    The sequence of real parts αn (resp. the absolute value) of T n (0) is exponentially increasing, that is, α n+1 ≥ cexp α n , for all \(n \in \mathbb{N}\) and for some c > 0.

  5. 5.

    That is there exist a > 0 and λ > 1 such that for all x, x′ ∈ π − 1({y}) we have that d(x, x′) < a implies that d(T(x), T(x′)) ≥ λd(x, x′).

  6. 6.

    That is for ε > 0 there exists some n ≥ 1 such that Tn (B(x,ε)) ⊃ π −1 ({S n(π(x))}).

  7. 7.

    That is ∑ \nolimits n=1 ∞ V n (ϕ) < ∞, where V n (ϕ) denotes the maximal variation of ϕ over cylinders of length n.

  8. 8.

    V n y (ϕ) ≤ κ(y)r n for n ≥ 2 and ∫ \nolimits \nolimits log κ d P < ∞.

  9. 9.

    For a fixed measurable family ξ y ∈ π −1 (y), we have that \({\sum \nolimits }_{n:{S}^{n}(y)\in Y ^{\prime}}{s}^{n}{({\mathcal{L}}_{\phi }^{(y)})}^{n}(1)({\xi }_{{S}^{n}(y)})\) converges for s < 1 and diverges for s = 1, where Y ′ is some set of positive measure.

  10. 10.

    That is for all ε > 0 there exists a measurable set K ⊂ X such that K ∩ π − 1({y}) is compact, for all y ∈ Y, and inf n ∫μ y (n)(K) dP(y) > 1 − ε. 

References

  1. Aaronson, J.; Denker, M.; Urbański, M.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.

    Google Scholar 

  2. Aaronson, J.; Denker, M.: The Poincaré series of \(\mathbb{C}\setminus \mathbb{Z}\). Ergodic Theory Dynam. Systems 19, No.1 (1999), 1–20.

    Google Scholar 

  3. Aaronson, J.; Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stochastics and Dynamics 1 (2001), 193–237.

    Google Scholar 

  4. Adachi, T.; Sunada, T.: Homology of closed geodesics in a negatively curved manifold. J. Differ. Geom. 26 (1987), 81–99.

    Google Scholar 

  5. Agol, I.: Tameness of hyperbolic 3-manifolds. Preprint (2004); arXiv:math.GT/0405568.

    Google Scholar 

  6. Avila, A.; Lyubich, M.: Hausdorff dimension and conformal measures of Feigenbaum Julia sets. J. Amer. Math. Soc. 21 (2008), 305–363.

    Google Scholar 

  7. Birkhoff, G.: Lattice theory. Third edition. AMS Colloquium Publ., Vol. XXV AMS, Providence, R.I. 1967

    Google Scholar 

  8. Bishop, C. J.; Jones, P. W.: Hausdorff dimension and Kleinian groups. Acta Math. 56 (1997), 1–39.

    Google Scholar 

  9. Blokh, A. M.; Mayer, J. C.; Oversteegen, L. G.: Recurrent critical points and typical limit sets for conformal measures. Topology Appl. 108 (2000), 233–244.

    Google Scholar 

  10. Bogenschütz, T.; Gundlach, M.: Ruelle’s transfer operator for random subshifts of finite type. Ergodic Theory Dynam. Systems 15 (1995), 413–447.

    Google Scholar 

  11. Bonfert-Taylor, P.; Matsuzaki, K.; Taylor, E. C.: Large and small covers of hyperbolic manifolds. J. Geom. Anal. 21 (2011) DOI: 10.1007/s12220-010-9204-6.

    Google Scholar 

  12. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes in Math. 470, Springer-Verlag, 1975.

    Google Scholar 

  13. Bowen, R.: Dimension of quasi circles. Publ. IHES 50 (1980), 11–25.

    Google Scholar 

  14. Brooks, R.: The bottom of the spectrum of a Riemannian cover. J. Reine Angew. Math. 357 (1985), 101–114.

    MathSciNet  MATH  Google Scholar 

  15. Bruin, H.; Todd, M.: Markov extensions and lifting measures for complex polynomials. Ergodic Theory Dynam. Systems 27 (2007), 743–768.

    Google Scholar 

  16. Buzzi, J.; Hubert, P.: Piecewise monotone maps without periodic points: rigidity, measures and complexity. Ergodic Theory Dynam. Systems 24 (2004), 383–405.

    Google Scholar 

  17. Buzzi, J.; Paccaut, F.; Schmitt, B.: Conformal measures for multidimensional piecewise invertible maps. Ergodic Theory Dynam. Systems 21 (2001), 1035–1049.

    Google Scholar 

  18. Calegari, D.; Gabai, D.: Shrinkwrapping and the taming of hyperbolic manifolds. J. Amer. Math. Soc. 19 (2006), 385–446.

    Google Scholar 

  19. Cornfeld, I. P.; Fomin, S. V.; Sinai, Ya. G.: Ergodic theory. Springer Verlag New York, Heidelberg, Berlin 1982.

    Google Scholar 

  20. Coven, E. M.; Reddy, W. L.: Positively expansive maps of compact manifolds. Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 96–110, Lecture Notes in Math., 819, Springer, Berlin, 1980.

    Google Scholar 

  21. Crauel, H.: Random probability measures on Polish spaces. Stochastics Monographs 11, Taylor & Francis, London, 2002.

    Google Scholar 

  22. Denker, M.; Gordin, M.: Gibbs measures for fibred systems. Adv. Math. 148 (1999), 161–192.

    Google Scholar 

  23. Denker, M.; Gordin, M.; Heinemann, S.-M.: On the relative variational principle for fibre expanding maps. Ergodic Theory Dynam. Systems 22 (2002), 757–782.

    Google Scholar 

  24. Denker, M.; Kifer, Y.; Stadlbauer, M.: Thermodynamic formalism for random countable Markov shifts. Discrete Contin. Dyn. Syst. 22 (2008), 131–164.

    Google Scholar 

  25. Denker, M.; Nitecki, Z.; Urbański, M.: Conformal measures and S-unimodal maps. Dynamical systems and applications, 169–212, World Sci. Ser. Appl. Anal., 4, World Sci. Publ., River Edge, NJ, 1995.

    Google Scholar 

  26. Denker, M.; Mauldin, R. D.; Nitecki, Z.; Urbański, M.: Conformal measures for rational functions revisited. Dedicated to the memory of Wieslaw Szlenk. Fund. Math. 157 (1998), 161–173.

    Google Scholar 

  27. Denker, M.; Przytycki, F.; Urbański, M.: On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory Dynam. Systems 16 (1996), 255–266.

    Google Scholar 

  28. Denker, M.; Urbański, M.: On the existence of conformal measures. Trans. Amer. Math. Soc. 328 (1991), 563–587.

    Google Scholar 

  29. Denker, M.; Urbański, M.: Hausdorff measures on Julia sets of subexpanding rational maps. Israel J. Math. 76 (1991), 193–214.

    Google Scholar 

  30. Denker, M.; Urbański, M.: Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points. Forum Math. 3 (1991), 561–579.

    Google Scholar 

  31. Denker, M.; Urbański, M.: On Sullivan’s conformal measures for rational maps of the Riemann sphere. Nonlinearity 4 (1991), 365–384.

    Google Scholar 

  32. Denker, M.; Urbański, M.: Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point. J. London Math. Soc. (2) 43 (1991), 107–118.

    Google Scholar 

  33. Denker, M.; Urbański, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4 (1991), 103–134.

    Google Scholar 

  34. Denker, M.; Urbański, M.: Geometric measures for parabolic rational maps. Ergodic Theory Dynam. Systems 12 (1992), 53–66.

    Google Scholar 

  35. Denker, M.; Yuri, M.: A note on the construction of nonsingular Gibbs measures. Colloquium Mathematicum 84/85 (2000), 377–383.

    Google Scholar 

  36. Douady, A.; Sentenac, P.; Zinsmeister, M.: Implosion parabolique et dimension de Hausdorff. C.R. Acad. Sci. Paris Sér. I. 325 (1997), 765–772.

    Google Scholar 

  37. Falk, K. H.; Stratmann, B. O.: Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups. Tohoku Math. Jour. 56 (4) (2004), 571–582.

    Google Scholar 

  38. Ferrero, P.; Schmitt, B.: Ruelle’s Perron-Frobenius theorem and projective metrics. In Random Fields Esztergom Hung., 1979, Coll. Math. Soc. Janos Bolyai, 27, 333–336. North-Holland, Amsterdam, 1979.

    Google Scholar 

  39. Gelfert, K.; Rams, M.: Geometry of limit sets for expansive Markov systems. Trans. Amer. Math. Soc. 361 (2009), 2001–2020.

    Google Scholar 

  40. Graczyk, J.; Smirnov, S.: Non-uniform hyperbolicity in complex dynamics. Invent. Math. 175 (2008), 335–415.

    Google Scholar 

  41. Gundlach, M.; Kifer, Y.: Expansiveness, specification, and equilibrium states for random bundle transformations. Discrete and Continuous Dynam. Syst. 6 (2000), 89–120.

    Google Scholar 

  42. Hill, R.; Velani, S.:The Jarník-Besicovitch theorem for geometrically finite Kleinian groups. Proc. London Math. Soc. (3) 77 (1998), 524 - 550.

    Google Scholar 

  43. Hofbauer, F.: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval. Studia Math. 103 (1992), 191–206.

    MathSciNet  MATH  Google Scholar 

  44. Hofbauer, F.: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval. II. Studia Math. 106 (1993), 213–231.

    MathSciNet  MATH  Google Scholar 

  45. Hofbauer, F. Hausdorff and conformal measures for weakly expanding maps of the interval. Differential Equations Dynam. Systems 2 (1994), 121–136.

    MathSciNet  MATH  Google Scholar 

  46. Hofbauer, F.: The Rényi dimension of a conformal measure for a piecewise monotonic map of the interval. Acta Math. Hungar. 107 (2005), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  47. Hofbauer, F.; Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.

    Google Scholar 

  48. Huang, Z.; Jiang, Y.; Wang, Y.: On conformal measures for infinitely renormalizable quadratic polynomials. Sci. China Ser. A 48 (2005), 1411–1420.

    Google Scholar 

  49. Huang, Z.; Wang, Y. On the existence of conformal measures supported on conical points. Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), 154–157.

    Google Scholar 

  50. Ionesu-Tulcea, C.; Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètment continues. Ann. of Math. 52 (1950), 140–147.

    Google Scholar 

  51. Kaimanovich, V. A.; Lyubich, M.: Conformal and harmonic measures on laminations associated with rational maps. Mem. Amer. Math. Soc. 173 (2005) 119 pp.

    Google Scholar 

  52. Keane, M.: Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.

    Article  MathSciNet  MATH  Google Scholar 

  53. Kesseböhmer, M.; Stratmann, B. O.: A note on the algebraic growth rate of Poincaré series for Kleinian groups. Article dedicated to S.J. Patterson’s 60th birthday (2011), in this volume.

    Google Scholar 

  54. K. Khanin, Y. Kifer: Thermodynamics formalism for random transformations and statistical mechanics. Amer. Math. Soc. Translations, Series 2 (1995), 107–140.

    Google Scholar 

  55. Khintchine, A. Y.: Continued fractions. Univ. of Chicago Press, Chicago and London, 1964.

    MATH  Google Scholar 

  56. Kifer, Y.: Equilibrium states for random expanding transformations. Random Comput. Dynam. 1 (1992/93), 1–31.

    Google Scholar 

  57. Kotus, J.: Conformal measures for non-entire functions with critical values eventually mapped onto infinity. Analysis (Munich) 25 (2005), 333–350.

    Article  MathSciNet  Google Scholar 

  58. Kotus, J.: Probabilistic invariant measures for non-entire functions with asymptotic values mapped onto ∞. Illinois J. Math. 49 (2005), 1203–1220.

    Google Scholar 

  59. Kotus, J.: Elliptic functions with critical points eventually mapped onto infinity. Monatsh. Math. 149 (2006), 103–117.

    Article  MathSciNet  MATH  Google Scholar 

  60. Kotus, J.; Urbański, M.: Conformal, geometric and invariant measures for transcendental expanding functions. Math. Ann. 324 (2002), 619–656.

    Google Scholar 

  61. Kotus, J.; Urbański, M.: Geometry and ergodic theory of non-recurrent elliptic functions. J. Anal. Math. 93 (2004), 35–102.

    Google Scholar 

  62. Kotus, J.; Urbański, M.: Geometry and dynamics of some meromorphic functions. Math. Nachr. 279 (2006), 1565–1584.

    Google Scholar 

  63. Kotus, J.; Urbański, M.: Fractal measures and ergodic theory for transcendental meromorphic functions. London Math. Soc. Lecture Note Series 348 (2008), 251–316.

    Google Scholar 

  64. Lalley, S.: Closed geodesics in homology classes on surfaces of variable negative curvature. Duke Math. J. 58 (1989), 795–821.

    Article  MathSciNet  MATH  Google Scholar 

  65. Lyons, T. J.; McKean, H. P.: Winding of the plane Brownian motion. Adv. Math. 51 (1984), 212–225.

    Google Scholar 

  66. Lyubich, M.: Entropy properties of rational endomorphismsof the Riemann sphere. Ergodic Theory and Dynam. Syst. 3 (1983), 351–386.

    MathSciNet  MATH  Google Scholar 

  67. Lyubich, M.: Dynamics of quadratic polynomials. I, II. Acta Math. 178 (1997), 185–247, 247–297.

    Google Scholar 

  68. Manning, A.; McCluskey, H.: Hausdorff dimension for horseshoes. Ergodic Theory Dynam. Systems 5 (1985), 71–88.

    Google Scholar 

  69. Mauldin, R. D.; Urbański, M.: Gibbs states on the symbolic space over an innite alphabet. Israel J. Maths. 125 (2001), 93–130.

    Google Scholar 

  70. Mauldin, R. D.; Urbański, M.: Graph directed Markov systems. Geometry and dynamics of limit sets. Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003.

    Google Scholar 

  71. Mayer, V.: Comparing measures and invariant line fields. Ergodic Theory Dynam. Systems 22 (2002), 555–570.

    Article  MathSciNet  MATH  Google Scholar 

  72. McKean, H. P.; Sullivan, D.: Brownian motion and harmonic functions on the class surface of the thrice punctured torus. Adv. Math. 51 (1984), 203–211.

    Google Scholar 

  73. Misiurewicz, M.: A short proof of the variational principle for a \({\mathbb{Z}}_{+}^{n}\) action on a compact space. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 1069–1075.

    MathSciNet  Google Scholar 

  74. Parry, W.: Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.

    Article  MathSciNet  MATH  Google Scholar 

  75. Patterson, S. J.: The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.

    Article  MathSciNet  MATH  Google Scholar 

  76. Patterson, S. J.: Diophantine approximation in Fuchsian groups. Phil. Trans. Roy. Soc. Lond. 282 (1976), 527–563.

    Article  MathSciNet  MATH  Google Scholar 

  77. Patterson, S. J.: Further remarks on the exponent of convergence of Poincaré series. Tôhoku Math. Journ. 35 (1983), 357–373.

    Article  MathSciNet  MATH  Google Scholar 

  78. Patterson, S. J.: Lectures on measures on limit sets of Kleinian groups. In Analytical and geometric aspects of hyperbolic space, (eds. D. B. A. Epstein); Cambridge University Press, 1987.

    Google Scholar 

  79. Phillips, R.; Sarnak, P.: Geodesics in homology classes. Duke Math. J. 55 (1987), 287–297.

    Google Scholar 

  80. Pollicott, M.; Sharp, R.: Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature. Invent. Math. 117 (1994), 275–302.

    Google Scholar 

  81. Prado, E. A. Ergodicity of conformal measures for unimodal polynomials. Conform. Geom. Dyn. 2 (1998), 29–44.

    Article  MathSciNet  MATH  Google Scholar 

  82. Prado, E. A.: Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials. Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), 31–52.

    Google Scholar 

  83. Przytycki, F.: Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119 (1993), 309–317.

    Article  MathSciNet  MATH  Google Scholar 

  84. Przytycki, F.: Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials. Trans. Amer. Math. Soc. 350 (1998), 717–742.

    Article  MathSciNet  MATH  Google Scholar 

  85. Przytycki, F.: Conical limit set and Poincaré exponent for iterations of rational functions. Trans. Amer. Math. Soc. 351 (1999), 2081–2099.

    Article  MathSciNet  MATH  Google Scholar 

  86. Rees, M.: Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergodic Theory Dynam. Systems 1 (1981), 107–133.

    MathSciNet  MATH  Google Scholar 

  87. Rees, M.: Divergence type of some subgroups of finitely generated Fuchsian groups. Ergodic Theory Dynam. Systems 1 (1981), 209–221.

    MathSciNet  MATH  Google Scholar 

  88. Rokhlin, V. A.: On the fundamental ideas of measure theory. Mat. Sbornik 25 (1949), 107-150. Transl.: Amer. Math. Soc. Translations 71 (1952), 1–54.

    Google Scholar 

  89. Roy, M.; Urbański, M.: Conformal families of measures for fibred systems. Monatsh. Math. 140 (2003), 135–145.

    Google Scholar 

  90. Rudolph, D. J.: Ergodic behaviour of Sullivan’s geometric mesure on a geometrically finite hyperbolic manifold. Ergodic Theory Dynam. Systems 2 (1982), 491–512.

    MathSciNet  MATH  Google Scholar 

  91. Sarig, O.: Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dynam. Systems 19 (1999), 1565–1593.

    Article  MathSciNet  MATH  Google Scholar 

  92. Sarig, O.: Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121 (2001), 285–311.

    Article  MathSciNet  MATH  Google Scholar 

  93. Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131 (2003), 1751–1758.

    Article  MathSciNet  MATH  Google Scholar 

  94. Skorulski, B.: The existence of conformal measures for some transcendental meromorphic functions. Complex dynamics, 169–201, Contemp. Math. 396, Amer. Math. Soc., Providence, RI, 2006.

    Google Scholar 

  95. Stadlbauer, M.: On random topological Markov chains with big images and preimages. Stochastics & Dynamics 10 (2010), 77–95.

    Article  MathSciNet  MATH  Google Scholar 

  96. Stratmann, B. O.: Diophantine approximation in Kleinian groups. Math. Proc. Camb. Phil. Soc. 116 (1994), 57–78.

    Article  MathSciNet  MATH  Google Scholar 

  97. Stratmann, B. O.: A note on counting cuspidal excursions. Annal. Acad. Sci. Fenn. 20 (1995), 359–372.

    MathSciNet  MATH  Google Scholar 

  98. Stratmann, B. O.: Fractal dimensions for Jarník limit sets; the semi-classical approach. Ark. för Mat. 33 (1995), 385–403.

    Article  MathSciNet  MATH  Google Scholar 

  99. Stratmann, B. O.: The Hausdorff dimension of bounded geodesics on geometrically finite manifolds. Ergodic Theory Dynam. Systems 17 (1997), 227–246.

    Article  MathSciNet  MATH  Google Scholar 

  100. Stratmann, B. O.: A remark on Myrberg initial data for Kleinian groups. Geometriae Dedicata 65 (1997), 257–266.

    Article  MathSciNet  MATH  Google Scholar 

  101. Stratmann, B. O.: Multiple fractal aspects of conformal measures; a survey. In Workshop on Fractals and Dynamics, (eds. M. Denker, S.-M. Heinemann, B. O. Stratmann); Math. Gottingensis 05 (1997), 65–71.

    Google Scholar 

  102. Stratmann, B. O.: Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements. Michigan Math. Jour. 46 (1999), 573–587.

    Article  MathSciNet  MATH  Google Scholar 

  103. Stratmann, B. O.: The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones. Progress in Probability 57 (2004), 93–107.

    MathSciNet  Google Scholar 

  104. Stratmann, B. O.: Fractal geometry on hyperbolic manifolds. In Non-Euclidean Geometries, Janos Bolyai Memorial Volume, (eds. A. Prekopa and E. Molnar); Springer Verlag, (2006), 227–249.

    Google Scholar 

  105. Stratmann, B. O.; Urbański, M.: The box-counting dimension for geometrically finite Kleinian groups. Fund. Matem. 149 (1996), 83–93.

    Google Scholar 

  106. Stratmann, B. O.; Urbański, M.: The geometry of conformal measures for parabolic rational maps. Math. Proc. Cambridge Philos. Soc. 128 (2000), 141–156.

    Google Scholar 

  107. Stratmann, B. O.; Urbański, M.: Jarnik and Julia: a Diophantine analysis for parabolic rational maps. Math. Scand. 91 (2002), 27–54.

    Google Scholar 

  108. Stratmann, B. O.; Urbański, M.: Metrical Diophantine analysis for tame parabolic iterated function systems. Pacific J. Math. 216 (2004), 361–392.

    Google Scholar 

  109. Stratmann, B. O.; Urbański, M.: Pseudo-Markov systems and infinitely generated Schottky groups. Amer. J. Math. 129 (4) (2007), 1019–1063.

    Google Scholar 

  110. Stratmann, B. O.; Velani, S.: The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. Lond. Math. Soc. (3) 71 (1995), 197–220.

    Google Scholar 

  111. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. IHES Publ. Math.50 (1979), 171–202.

    MATH  Google Scholar 

  112. Sullivan, D.: Conformal dynamical systems. In Geometric Dynamics; Lect. Notes in Math. 1007 (1983), 725–752.

    Google Scholar 

  113. Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.

    Article  MathSciNet  MATH  Google Scholar 

  114. Sumi, H.: On dynamics of hyperbolic rational semigroups and Hausdorff dimension of Julia sets. In Complex dynamical systems and related areas; (Japanese) (Kyoto, 1996). Surikaisekikenkyusho Kokyuroku 988 (1997), 98–112.

    Google Scholar 

  115. Sumi, H.: Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and conformal measures of rational semigroups. In Problems on complex dynamical systems; (Japanese) (Kyoto, 1997). Surikaisekikenkyusho Kokyuroku 1042 (1998), 68–77.

    Google Scholar 

  116. Sumi, H.: On Hausdorff dimension of Julia sets of hyperbolic rational semigroups. Kodai Math. Jour. 21 (1998), 10–28.

    Article  MathSciNet  MATH  Google Scholar 

  117. Urbański, M.: Measures and dimensions in conformal dynamics. Bull. Amer. Math. Soc. (N.S.) 40 (2003), 281–321

    Google Scholar 

  118. Urbański, M.: Geometry and ergodic theory of conformal non-recurrent dynamics. Ergodic Theory Dynam. Systems 17 (1997), 1449–1476.

    Article  MathSciNet  MATH  Google Scholar 

  119. Urbański, M.; Zdunik, A.: The finer geometry and dynamics of the hyperbolic exponential family. Michigan Math. J. 51 (2003), 227–250.

    Google Scholar 

  120. Urbański, M.; Zdunik, A.: The parabolic map \({f}_{1/e}(z) = \frac{1} {e}{e}^{z}\). Indag. Math. (N.S.) 15 (2004), 419–433.

    Google Scholar 

  121. Urbański, M.; Zdunik, A.: Geometry and ergodic theory of non-hyperbolic exponential maps. Trans. Amer. Math. Soc. 359 (2007), 3973–3997.

    Google Scholar 

  122. Walters, P.: A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975), 937–971.

    Article  MathSciNet  Google Scholar 

  123. Yuri, M.: On the construction of conformal measures for piecewise C 0-invertible systems. In Studies on complex dynamics and related topics; (Japanese) (Kyoto, 2000). Surikaisekikenkyusho Kokyuroku 1220 (2001), 141–143.

    Google Scholar 

  124. Yuri, M.: Thermodynamic Formalism for countable to on Markov systems. Trans. Amer. Math. Soc. 355 (2003), 2949–2971.

    Article  MathSciNet  MATH  Google Scholar 

  125. Zinsmeister, M.: Formalisme thermodynamique et systèmes dynamiques holomorphes. (French) [Thermodynamical formalism and holomorphic dynamical systems] Panoramas et Synthéses [Panoramas and Syntheses], 4. Société Mathématique de France, Paris, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Pennsylvania State University, State College, PA, 16802, USA

    Manfred Denker

  2. Fachbereich 3 – Mathematik und Informatik, Universität Bremen, Bibliothekstr. 1, 28359, Bremen, Germany

    Bernd O. Stratmann

Authors
  1. Manfred Denker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernd O. Stratmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manfred Denker .

Editor information

Editors and Affiliations

  1. , Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, Göttingen, 37073, Germany

    Valentin Blomer

  2. Mathematisches Institut, Universität Göttingen, Göttingen, 37073, Germany

    Preda Mihăilescu

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this paper

Cite this paper

Denker, M., Stratmann, B.O. (2012). The Patterson Measure: Classics, Variations and Applications. In: Blomer, V., Mihăilescu, P. (eds) Contributions in Analytic and Algebraic Number Theory. Springer Proceedings in Mathematics, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1219-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation