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Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order

Published online by Cambridge University Press:  01 June 2008

VOLKER MAYER  and
MARIUSZ URBAŃSKI
VOLKER MAYER
Affiliation:
Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France (email: [email protected])
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA (email: [email protected])
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Abstract

Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 915 - 946
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Barański, K.. Hausdorff dimension and measures on Julia sets of some meromorphic functions. Fund. Math. 147 (1995), 239–260.CrossRefGoogle Scholar
[2]Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29(2) (1993), 151–188.CrossRefGoogle Scholar
[3]Berteloot, F. and Mayer, V.. Rudiments de Dynamique Holomorphe (Cours Spécialisés, 7). Soc. Math. France, Paris, 2000.Google Scholar
[4]Borel, É.. Sur les zéros des fonctions entières. Acta Math. 20 (1897), 357–396.CrossRefGoogle Scholar
[5]Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. Inst. Hautes Études Sci. 50 (1980), 11–25.CrossRefGoogle Scholar
[6]Coiculescu, I. and Skorulski, B.. Thermodynamic formalism of transcendental entire maps of finite singular type. Monatsh. Math. 152(2) (2007), 105–123.CrossRefGoogle Scholar
[7]Coiculescu, I. and Skorulski, B.. Perturbations in the Speiser class. Rocky Mountain J. Math. 37(3) (2007), 763–800.CrossRefGoogle Scholar
[8]Cherry, W. and Ye, Z.. Nevanlinna’s Theory of Value Distribution (Springer Monographs in Mathematics). Springer, Berlin, 2001.CrossRefGoogle Scholar
[9]Denker, M. and Urbański, M.. On the existence of conformal measures. Trans. Amer. Math. Soc. 328 (1991), 563–587.CrossRefGoogle Scholar
[10]Denker, M. and Urbański, M.. Ergodic theory of equilibrium states. Nonlinearity 4 (1991), 103–134.CrossRefGoogle Scholar
[11]Eremenko, A.. On the iterations of entire functions. Dynamical Systems and the Ergodic Theory (Banach Center Publication, 23). Polish Academy Science, Warsaw, 1989.Google Scholar
[12]Eremenko, A.. Ahlfors’ contribution to the theory of meromorphic functions.Google Scholar
[13]Eremenko, A. E. and Lyubich, M. Yu.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier, Grenoble 42(4) (1992), 989–1020.CrossRefGoogle Scholar
[14]Hille, E.. Analytic Function Theory, Vol. II. Ginn, Boston, MA, 1962.Google Scholar
[15]Hinkkanen, A.. A sharp form of Nevanlinna’s second fundamental theorem. Invent. Math. 108 (1992), 549–574.CrossRefGoogle Scholar
[16]Ionescu-Tulcea, C. and Marinescu, G.. Théorie ergodique pour des classes d’operations non-complètement continues. Ann. Math. 52 (1950), 140–147.CrossRefGoogle Scholar
[17]Iversen, F.. Recherches sur les fonctions inverses des fonctions méromorphes. Thèse de Helsingfors, 1914.Google Scholar
[18]Kato, T.. Perturbation Theory for Linear Operators. Springer, Berlin, 1995.CrossRefGoogle Scholar
[19]Kotus, J. and Urbański, M.. Conformal, geometric and invariant measures for transcendental expanding functions. Math. Ann. 324 (2002), 619–656.CrossRefGoogle Scholar
[20]Kotus, J. and Urbański, M.. Geometry and ergodic theory of non-recurrent elliptic functions. J. Anal. Math. 93 (2004), 35–102.CrossRefGoogle Scholar
[21]Kotus, J. and Urbański, M.. The dynamics and geometry of the Fatou functions. Discrete Contin. Dyn. Sys. 13 (2005), 291–338.CrossRefGoogle Scholar
[22]Kotus, J. and Urbański, M.. Fractal measures and ergodic theory of transcendental meromorphic functions. Preprint, 2004. London Mathematical Society Lecture Notes, to appear.Google Scholar
[23]Lyubich, M. Yu.. Some typical properties of the dynamics of rational maps. Russian Math. Surveys 8(5) (1983), 154–155.CrossRefGoogle Scholar
[24]Lyubich, M. Yu.. The dynamics of rational transforms: the topological picture. Russian Math. Surveys 41(4) (1986), 43–117.CrossRefGoogle Scholar
[25]Mañé, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. École. Norm. Sup. 4 16 (1983), 193–217.CrossRefGoogle Scholar
[26]McMullen, C.. Area and Hausdorff dimension of Julia set of entire functions. Trans. Amer. Math. Soc. 300 (1987), 329–342.CrossRefGoogle Scholar
[27]Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces (Cambridge Studies in Advanced Mathematics, 44). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[28]Mauldin, D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. London Math. Soc. (3) 73 (1996), 105–154.CrossRefGoogle Scholar
[29]Mayer, V.. Rational functions without conformal measures on the conical set. Preprint, 2002.Google Scholar
[30]Mayer, V.. The size of the Julia set of meromorphic functions. Preprint, 2005.Google Scholar
[31]Mayer, V. and Urbański, M.. Gibbs and equilibrium measures for elliptic functions. Math. Z. 250 (2005), 657–683.CrossRefGoogle Scholar
[32]Nevanlinna, R.. Eindeutige Analytische Funktionen. Springer, Berlin, 1953.CrossRefGoogle Scholar
[33]Nevanlinna, R.. Analytic Functions. Springer, Berlin, 1970.CrossRefGoogle Scholar
[34]Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.CrossRefGoogle Scholar
[35]Przytycki, F. and Urbański, M.. Fractals in the Plane — the Ergodic Theory Methods, available on Urbański’s webpage; Cambridge University Press, to appear.Google Scholar
[36]Rippon, P. J. and Stallard, G. M.. Iteration of a class of hyperbolic meromorphic functions. Proc. Amer. Math. Soc. 127(11) (1999), 3251–3258.CrossRefGoogle Scholar
[37]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.CrossRefGoogle Scholar
[38]Slodkowski, Z.. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 347–355.CrossRefGoogle Scholar
[39]Sullivan, D.. Seminar on conformal and hyperbolic geometry. Preprint, IHES, 1982.Google Scholar
[40]Urbański, M. and Zdunik, A.. The finer geometry and dynamics of exponential family. Michigan Math. J. 51 (2003), 227–250.CrossRefGoogle Scholar
[41]Urbański, M. and Zdunik, A.. Real analyticity of Hausdorff dimension of finer Julia sets of exponential family. Ergod. Th. & Dynam. Sys. 24 (2004), 279–315.CrossRefGoogle Scholar
[42]Urbański, M. and Zdunik, A.. Geometry and ergodic theory of non-hyperbolic exponential maps. Trans. Amer. Math. Soc. 359 (2007), 3973–3997.CrossRefGoogle Scholar