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

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.