The Patterson Measure: Classics, Variations and Applications

Conference paper
Part of the Springer Proceedings in Mathematics book series (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.

References

  1. 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. 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. 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. 4.
    Adachi, T.; Sunada, T.: Homology of closed geodesics in a negatively curved manifold. J. Differ. Geom. 26 (1987), 81–99.Google Scholar
  5. 5.
    Agol, I.: Tameness of hyperbolic 3-manifolds. Preprint (2004); arXiv:math.GT/0405568.Google Scholar
  6. 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. 7.
    Birkhoff, G.: Lattice theory. Third edition. AMS Colloquium Publ., Vol. XXV AMS, Providence, R.I. 1967Google Scholar
  8. 8.
    Bishop, C. J.; Jones, P. W.: Hausdorff dimension and Kleinian groups. Acta Math. 56 (1997), 1–39.Google Scholar
  9. 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. 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. 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. 12.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes in Math. 470, Springer-Verlag, 1975.Google Scholar
  13. 13.
    Bowen, R.: Dimension of quasi circles. Publ. IHES 50 (1980), 11–25.Google Scholar
  14. 14.
    Brooks, R.: The bottom of the spectrum of a Riemannian cover. J. Reine Angew. Math. 357 (1985), 101–114.MathSciNetMATHGoogle Scholar
  15. 15.
    Bruin, H.; Todd, M.: Markov extensions and lifting measures for complex polynomials. Ergodic Theory Dynam. Systems 27 (2007), 743–768.Google Scholar
  16. 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. 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. 18.
    Calegari, D.; Gabai, D.: Shrinkwrapping and the taming of hyperbolic manifolds. J. Amer. Math. Soc. 19 (2006), 385–446.Google Scholar
  19. 19.
    Cornfeld, I. P.; Fomin, S. V.; Sinai, Ya. G.: Ergodic theory. Springer Verlag New York, Heidelberg, Berlin 1982.Google Scholar
  20. 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. 21.
    Crauel, H.: Random probability measures on Polish spaces. Stochastics Monographs 11, Taylor & Francis, London, 2002.Google Scholar
  22. 22.
    Denker, M.; Gordin, M.: Gibbs measures for fibred systems. Adv. Math. 148 (1999), 161–192.Google Scholar
  23. 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. 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. 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. 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. 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. 28.
    Denker, M.; Urbański, M.: On the existence of conformal measures. Trans. Amer. Math. Soc. 328 (1991), 563–587.Google Scholar
  29. 29.
    Denker, M.; Urbański, M.: Hausdorff measures on Julia sets of subexpanding rational maps. Israel J. Math. 76 (1991), 193–214.Google Scholar
  30. 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. 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. 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. 33.
    Denker, M.; Urbański, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4 (1991), 103–134.Google Scholar
  34. 34.
    Denker, M.; Urbański, M.: Geometric measures for parabolic rational maps. Ergodic Theory Dynam. Systems 12 (1992), 53–66.Google Scholar
  35. 35.
    Denker, M.; Yuri, M.: A note on the construction of nonsingular Gibbs measures. Colloquium Mathematicum 84/85 (2000), 377–383.Google Scholar
  36. 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. 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. 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. 39.
    Gelfert, K.; Rams, M.: Geometry of limit sets for expansive Markov systems. Trans. Amer. Math. Soc. 361 (2009), 2001–2020.Google Scholar
  40. 40.
    Graczyk, J.; Smirnov, S.: Non-uniform hyperbolicity in complex dynamics. Invent. Math. 175 (2008), 335–415.Google Scholar
  41. 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. 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. 43.
    Hofbauer, F.: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval. Studia Math. 103 (1992), 191–206.MathSciNetMATHGoogle Scholar
  44. 44.
    Hofbauer, F.: Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval. II. Studia Math. 106 (1993), 213–231.MathSciNetMATHGoogle Scholar
  45. 45.
    Hofbauer, F. Hausdorff and conformal measures for weakly expanding maps of the interval. Differential Equations Dynam. Systems 2 (1994), 121–136.MathSciNetMATHGoogle Scholar
  46. 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.MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Hofbauer, F.; Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.Google Scholar
  48. 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. 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. 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. 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. 52.
    Keane, M.: Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.MathSciNetMATHCrossRefGoogle Scholar
  53. 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. 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. 55.
    Khintchine, A. Y.: Continued fractions. Univ. of Chicago Press, Chicago and London, 1964.MATHGoogle Scholar
  56. 56.
    Kifer, Y.: Equilibrium states for random expanding transformations. Random Comput. Dynam. 1 (1992/93), 1–31.Google Scholar
  57. 57.
    Kotus, J.: Conformal measures for non-entire functions with critical values eventually mapped onto infinity. Analysis (Munich) 25 (2005), 333–350.MathSciNetCrossRefGoogle Scholar
  58. 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. 59.
    Kotus, J.: Elliptic functions with critical points eventually mapped onto infinity. Monatsh. Math. 149 (2006), 103–117.MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Kotus, J.; Urbański, M.: Conformal, geometric and invariant measures for transcendental expanding functions. Math. Ann. 324 (2002), 619–656.Google Scholar
  61. 61.
    Kotus, J.; Urbański, M.: Geometry and ergodic theory of non-recurrent elliptic functions. J. Anal. Math. 93 (2004), 35–102.Google Scholar
  62. 62.
    Kotus, J.; Urbański, M.: Geometry and dynamics of some meromorphic functions. Math. Nachr. 279 (2006), 1565–1584.Google Scholar
  63. 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. 64.
    Lalley, S.: Closed geodesics in homology classes on surfaces of variable negative curvature. Duke Math. J. 58 (1989), 795–821.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Lyons, T. J.; McKean, H. P.: Winding of the plane Brownian motion. Adv. Math. 51 (1984), 212–225.Google Scholar
  66. 66.
    Lyubich, M.: Entropy properties of rational endomorphismsof the Riemann sphere. Ergodic Theory and Dynam. Syst. 3 (1983), 351–386.MathSciNetMATHGoogle Scholar
  67. 67.
    Lyubich, M.: Dynamics of quadratic polynomials. I, II. Acta Math. 178 (1997), 185–247, 247–297.Google Scholar
  68. 68.
    Manning, A.; McCluskey, H.: Hausdorff dimension for horseshoes. Ergodic Theory Dynam. Systems 5 (1985), 71–88.Google Scholar
  69. 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. 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. 71.
    Mayer, V.: Comparing measures and invariant line fields. Ergodic Theory Dynam. Systems 22 (2002), 555–570.MathSciNetMATHCrossRefGoogle Scholar
  72. 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. 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.MathSciNetGoogle Scholar
  74. 74.
    Parry, W.: Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Patterson, S. J.: The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    Patterson, S. J.: Diophantine approximation in Fuchsian groups. Phil. Trans. Roy. Soc. Lond. 282 (1976), 527–563.MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    Patterson, S. J.: Further remarks on the exponent of convergence of Poincaré series. Tôhoku Math. Journ. 35 (1983), 357–373.MathSciNetMATHCrossRefGoogle Scholar
  78. 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. 79.
    Phillips, R.; Sarnak, P.: Geodesics in homology classes. Duke Math. J. 55 (1987), 287–297.Google Scholar
  80. 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. 81.
    Prado, E. A. Ergodicity of conformal measures for unimodal polynomials. Conform. Geom. Dyn. 2 (1998), 29–44.MathSciNetMATHCrossRefGoogle Scholar
  82. 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. 83.
    Przytycki, F.: Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119 (1993), 309–317.MathSciNetMATHCrossRefGoogle Scholar
  84. 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.MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    Przytycki, F.: Conical limit set and Poincaré exponent for iterations of rational functions. Trans. Amer. Math. Soc. 351 (1999), 2081–2099.MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Rees, M.: Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergodic Theory Dynam. Systems 1 (1981), 107–133.MathSciNetMATHGoogle Scholar
  87. 87.
    Rees, M.: Divergence type of some subgroups of finitely generated Fuchsian groups. Ergodic Theory Dynam. Systems 1 (1981), 209–221.MathSciNetMATHGoogle Scholar
  88. 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. 89.
    Roy, M.; Urbański, M.: Conformal families of measures for fibred systems. Monatsh. Math. 140 (2003), 135–145.Google Scholar
  90. 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.MathSciNetMATHGoogle Scholar
  91. 91.
    Sarig, O.: Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dynam. Systems 19 (1999), 1565–1593.MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    Sarig, O.: Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121 (2001), 285–311.MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131 (2003), 1751–1758.MathSciNetMATHCrossRefGoogle Scholar
  94. 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. 95.
    Stadlbauer, M.: On random topological Markov chains with big images and preimages. Stochastics & Dynamics 10 (2010), 77–95.MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Stratmann, B. O.: Diophantine approximation in Kleinian groups. Math. Proc. Camb. Phil. Soc. 116 (1994), 57–78.MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Stratmann, B. O.: A note on counting cuspidal excursions. Annal. Acad. Sci. Fenn. 20 (1995), 359–372.MathSciNetMATHGoogle Scholar
  98. 98.
    Stratmann, B. O.: Fractal dimensions for Jarník limit sets; the semi-classical approach. Ark. för Mat. 33 (1995), 385–403.MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    Stratmann, B. O.: The Hausdorff dimension of bounded geodesics on geometrically finite manifolds. Ergodic Theory Dynam. Systems 17 (1997), 227–246.MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Stratmann, B. O.: A remark on Myrberg initial data for Kleinian groups. Geometriae Dedicata 65 (1997), 257–266.MathSciNetMATHCrossRefGoogle Scholar
  101. 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. 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.MathSciNetMATHCrossRefGoogle Scholar
  103. 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.MathSciNetGoogle Scholar
  104. 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. 105.
    Stratmann, B. O.; Urbański, M.: The box-counting dimension for geometrically finite Kleinian groups. Fund. Matem. 149 (1996), 83–93.Google Scholar
  106. 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. 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. 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. 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. 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. 111.
    Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. IHES Publ. Math.50 (1979), 171–202.MATHGoogle Scholar
  112. 112.
    Sullivan, D.: Conformal dynamical systems. In Geometric Dynamics; Lect. Notes in Math. 1007 (1983), 725–752.Google Scholar
  113. 113.
    Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259–277.MathSciNetMATHCrossRefGoogle Scholar
  114. 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. 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. 116.
    Sumi, H.: On Hausdorff dimension of Julia sets of hyperbolic rational semigroups. Kodai Math. Jour. 21 (1998), 10–28.MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    Urbański, M.: Measures and dimensions in conformal dynamics. Bull. Amer. Math. Soc. (N.S.) 40 (2003), 281–321Google Scholar
  118. 118.
    Urbański, M.: Geometry and ergodic theory of conformal non-recurrent dynamics. Ergodic Theory Dynam. Systems 17 (1997), 1449–1476.MathSciNetMATHCrossRefGoogle Scholar
  119. 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. 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. 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. 122.
    Walters, P.: A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975), 937–971.MathSciNetCrossRefGoogle Scholar
  123. 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. 124.
    Yuri, M.: Thermodynamic Formalism for countable to on Markov systems. Trans. Amer. Math. Soc. 355 (2003), 2949–2971.MathSciNetMATHCrossRefGoogle Scholar
  125. 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

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Authors and Affiliations

  1. 1.Mathematics DepartmentPennsylvania State UniversityState CollegeUSA
  2. 2.Fachbereich 3 – Mathematik und InformatikUniversität BremenBremenGermany

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