Non-recurrent meromorphic functions

Tom 182 / 2004

Jacek Graczyk, Janina Kotus, Grzegorz /Swiątek Fundamenta Mathematicae 182(2004), 269-281 MSC: Primary 37F50; Secondary 30D05. DOI: 10.4064/fm182-3-5


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.


  • Jacek GraczykDépartement de Mathématiques
    Université de Paris-Sud
    91405 Orsay, France
  • Janina KotusDepartment of Mathematics
    Warsaw University of Technology
    00-661 Warszawa, Poland
  • Grzegorz /SwiątekDepartment of Mathematics
    Penn State University
    University Park, PA 16802, U.S.A.

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