We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable invariant line-field.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-5,
author = {Jacek Graczyk and Janina Kotus and Grzegorz \'Swi\k atek},
title = {Non-recurrent meromorphic functions},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {269-281},
zbl = {1079.37041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-5}
}
Jacek Graczyk; Janina Kotus; Grzegorz Świątek. Non-recurrent meromorphic functions. Fundamenta Mathematicae, Tome 184 (2004) pp. 269-281. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-3-5/