Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T00:47:03.105Z Has data issue: false hasContentIssue false
  • English
  • Français

Bowen’s formula for meromorphic functions

Published online by Cambridge University Press:  13 June 2011

KRZYSZTOF BARAŃSKI ,
BOGUSŁAWA KARPIŃSKA  and
ANNA ZDUNIK
KRZYSZTOF BARAŃSKI
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland (email: [email protected], [email protected])
BOGUSŁAWA KARPIŃSKA
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland (email: [email protected])
ANNA ZDUNIK
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f be an arbitrary transcendental entire or meromorphic function in the class 𝒮 (i.e. with finitely many singularities). We show that the topological pressure P(f,t) for t>0 can be defined as the common value of the pressures P(f,t,z) for all z∈ℂ up to a set of Hausdorff dimension zero. Moreover, we prove that P(f,t) equals the supremum of the pressures of f∣X over all invariant hyperbolic subsets X of the Julia set, and we prove Bowen’s formula for f, i.e. we show that the Hausdorff dimension of the radial Julia set of f is equal to the infimum of the set of t, for which P(f,t) is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions f in the class ℬ (i.e. with a bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1165 - 1189
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barański, K.. Hausdorff dimension and measures on Julia sets of some meromorphic maps. Fund. Math. 147 (1995), 239–260.CrossRefGoogle Scholar
[2]Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151–188.CrossRefGoogle Scholar
[3]Bergweiler, W.. The role of the Ahlfors five islands theorem in complex dynamics. Conform. Geom. Dyn. 4 (2000), 22–34.CrossRefGoogle Scholar
[4]Bergweiler, W., Rippon, P. J. and Stallard, G. M.. Dynamics of meromorphic functions with direct or logarithmic singularities. Proc. Lond. Math. Soc. 97 (2008), 368–400.CrossRefGoogle Scholar
[5]Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.CrossRefGoogle Scholar
[6]Carleson, L. and Gamelin, T. W.. Complex Dynamics. Springer, New York, 1993.CrossRefGoogle Scholar
[7]Collingwood, E. F. and Lohwater, A. J.. The Theory of Cluster Sets. Cambridge University Press, London, 1966.CrossRefGoogle Scholar
[8]Eremenko, A. E. and Lyubich, M. Yu.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42 (1992), 989–1020.CrossRefGoogle Scholar
[9]Kotus, J. and Urbański, M.. Conformal, geometric and invariant measures for transcendental expanding functions. Math. Ann. 324 (2002), 619–656.CrossRefGoogle Scholar
[10]McMullen, C. T.. Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300 (1987), 329–342.CrossRefGoogle Scholar
[11]Mayer, V. and Urbański, M.. Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order. Ergod. Th. & Dynam. Sys. 28 (2008), 915–946.CrossRefGoogle Scholar
[12]Mayer, V. and Urbański, M.. Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order. Mem. Amer. Math. Soc. 954 (2010), 1–105.Google Scholar
[13]Przytycki, F.. Conical limit set and Poincaré exponent for iterations of rational functions. Trans. Amer. Math. Soc. 351 (1999), 2081–2099.CrossRefGoogle Scholar
[14]Przytycki, F., Rivera Letelier, J. and Smirnov, S.. Equality of pressures for rational functions. Ergod. Th. & Dynam. Sys. 24 (2004), 891–914.CrossRefGoogle Scholar
[15]Przytycki, F. and Urbański, M.. Conformal fractals. Ergodic Theory Methods (London Mathematical Society Lecture Note Series, 371). Cambridge University Press, Cambridge, 2010.Google Scholar
[16]Rempe, L.. Hyperbolic dimension and radial Julia sets of transcendental functions. Proc. Amer. Math. Soc. 137 (2008), 1411–1420.CrossRefGoogle Scholar
[17]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison-Wesley, Reading, MA, 1978.Google Scholar
[18]Stallard, G. M.. The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions. Math. Proc. Cambridge Philos. Soc. 127 (1999), 271–288.CrossRefGoogle Scholar
[19]Stallard, G. M.. The Hausdorff dimension of Julia sets of meromorphic functions. J. Lond. Math. Soc. 49 (1994), 281–295.CrossRefGoogle Scholar
[20]Urbański, M. and Zdunik, A.. The finer geometry and dynamics of the hyperbolic exponential family. Michigan Math. J. 51 (2003), 227–250.CrossRefGoogle Scholar
[21]Urbański, M. and Zdunik, A.. Real analyticity of Hausdorff dimension of the finer Julia sets of exponential family. Ergod. Th. & Dynam. Sys. 24 (2004), 279–315.CrossRefGoogle Scholar
[22]Urbański, M. and Zdunik, A.. Geometry and ergodic theory of non-hyperbolic exponential maps. Trans. Amer. Math. Soc. 359 (2007), 3973–3997.CrossRefGoogle Scholar
[23]Zheng, J.-H.. Dynamics of hyperbolic meromorphic functions. Preprint, 2008.Google Scholar