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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
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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 

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