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

TL;DR

This paper provides an elementary proof that invariant line fields do not exist on the conical Julia set for a broad class of analytic functions, including rational, transcendental meromorphic, and Ahlfors islands maps.

Contribution

It introduces a general proof technique for the absence of invariant differentials on conical Julia sets, extending previous results to Ahlfors islands maps.

Findings

No invariant line fields on conical Julia sets for these functions.

The proof applies to a very general setting, including Ahlfors islands maps.

Results have implications for rigidity questions in complex dynamics.

Abstract

In this note, 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 as well as transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Adam 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.