Non-recurrent meromorphic functions
Volume 182 / 2004
Fundamenta Mathematicae 182 (2004), 269-281
MSC: Primary 37F50; Secondary 30D05.
DOI: 10.4064/fm182-3-5
Abstract
We consider a transcendental meromorphic function $f$ belonging to the class ${\mathcal B}$ (with bounded set of singular values). We show that if the Julia set $J(f)$ is the whole complex plane ${\mathbb C}$, and the closure of the postcritical set $P(f)$ is contained in $B(0,R)\cup \{\infty \}$ 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.
