Non-recurrent meromorphic functions

Jacek Graczyk¹ ; Janina Kotus² ; Grzegorz /Swiątek³

¹Département de Mathématiques Université de Paris-Sud 91405 Orsay, France
²Department of Mathematics Warsaw University of Technology 00-661 Warszawa, Poland
³Department of Mathematics Penn State University University Park, PA 16802, U.S.A.

Fundamenta Mathematicae, Tome 182 (2004) no. 3, pp. 269-281

Résumé

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.

Détail

DOI : 10.4064/fm182-3-5

Keywords: consider transcendental meromorphic function belonging class mathcal bounded set singular values julia set whole complex plane mathbb closure postcritical set contained cup infty disjoint set crit critical points every compact forward invariant set hyperbolic provided disjoint crit further shown under general additional hypotheses admits measurable invariant line field