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Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

Published online by Cambridge University Press:  23 November 2011

VOLKER MAYER  and
LASSE REMPE
VOLKER MAYER
Affiliation:
Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France (email: volker.mayer@math.univ-lille1.fr)
LASSE REMPE
Affiliation:
Department of Mathematical Sciences, University of Liverpool, L69 7ZL, UK (email: l.rempe@liverpool.ac.uk)
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Abstract

In this paper, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational and transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Epstein. In fact, we prove a more general result on the absence of invariant differentials, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for log ∣f′∣, which are relevant to a number of well-known rigidity questions. In particular, we prove the absence of continuous line fields on the Julia set of any transcendental entire function.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1691 - 1710
Copyright
Copyright © Cambridge University Press 2011

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