  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.
