Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T17:10:26.127Z Has data issue: false hasContentIssue false
  • English
  • Français

Real analyticity for random dynamics of transcendental functions

Published online by Cambridge University Press:  10 August 2018

VOLKER MAYER ,
MARIUSZ URBAŃSKI  and
ANNA ZDUNIK
VOLKER MAYER
Affiliation:
Université de Lille, Département de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France email volker.mayer@univ-lille.fr
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA email urbanski@unt.edu
ANNA ZDUNIK
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland email A.Zdunik@mimuw.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh [On the dimension of conformal repellors, randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748] leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron–Frobenius operator are assumed to converge. We also provide Bowen’s formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application establishes real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 2 , February 2020 , pp. 490 - 520
Copyright
© Cambridge University Press, 2018 

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

Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.Google Scholar
Crauel, H.. Random Probability Measures on Polish Spaces (Stochastics Monographs, 11) . Taylor & Francis, London, 2002.Google Scholar
Daniel Mauldin, R. and Urbański, M.. Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets (Cambridge Tracts in Mathematics, 148) . Cambridge University Press, Cambridge, 2003.Google Scholar
Kifer, Y.. Thermodynamic formalism for random transformations revisited. Stoch. Dyn. 8 (2008), 77–102.Google Scholar
Mayer, V., Skorulski, B. and Urbanski, M.. Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry (Lecture Notes in Mathematics, 2036) . Springer, Heidelberg, 2011.Google Scholar
Mayer, V. and Urbański, M.. Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order. Ergod. Th. & Dynam. Sys. 28(3) (2008), 915–946.Google Scholar
Mayer, V. and Urbański, M.. Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order. Mem. Amer. Math. Soc. 203(954) (2010), 107 pp.Google Scholar
Mayer, V. and Urbański, M.. Random dynamics of transcendental functions. J. d’Anal. Math. 134 (2018), 201–235.Google Scholar
McMullen, C.. Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300(1) (1987), 329–342.Google Scholar
McMullen, C. T.. Complex Dynamics and Renormalization (Annals of Mathematics Studies, 135) . Princeton University Press, Princeton, NJ, 1994.Google Scholar
Pollicott, M.. Analyticity of dimensions for hyperbolic surface diffeomorphisms. Proc. Amer. Math. Soc. 143(8) (2015), 3465–3474.Google Scholar
Przytycki, F. and Urbański, M.. Conformal Fractals: Ergodic Theory Methods (London Mathematical Society Lecture Note Series, 371) . Cambridge University Press, Cambridge, 2010.Google Scholar
Ruelle, D.. Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics (Encyclopedia of Mathematics and its Applications, 5) . Addison-Wesley, Reading, MA, 1978, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota.Google Scholar
Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2(1) (1982), 99–107.Google Scholar
Rugh, H. H.. On the dimension of conformal repellors, randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748.Google Scholar
Rugh, H. H.. Cones and gauges in complex spaces: spectral gaps and complex Perron–Frobenius theory. Ann. of Math. (2) 171(3) (2010), 1707–1752.Google Scholar
Skorulski, B. and Urbański, M.. Finer fractal geometry for analytic families of conformal dynamical systems. Dyn. Syst. 29(3) (2014), 369–398.Google Scholar
Stallard, G. M.. The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions. II. Ergod. Th. & Dynam. Sys. 20(3) (2000), 895–910.Google Scholar
Urbański, M. and Zdunik, A.. The finer geometry and dynamics of the hyperbolic exponential family. Michigan Math. J. 51(2) (2003), 227–250.Google Scholar
Urbański, M. and Zdunik, A.. Real analyticity of Hausdorff dimension of finer Julia sets of exponential family. Ergod. Th. & Dynam. Sys. 24(1) (2004), 279–315.Google Scholar
Verjovsky, A. and Wu, H.. Hausdorff dimension of Julia sets of complex Hénon mappings. Ergod. Th. & Dynam. Sys. 16(4) (1996), 849–861.Google Scholar
Zinsmeister, M.. Thermodynamic Formalism and Holomorphic Dynamical Systems (SMF/AMS Texts and Monographs, 2) . American Mathematical Society, Providence, RI, 2000, Société Mathématique de France, Paris. Translated from the 1996 French original by C. Greg Anderson.Google Scholar