Normal families of light mappings of the sphere onto itself

V. V. Aseev ; D. G. Kuzin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 733-742

Résumé

Considering the class ${\mathcal D}$ of all continuous light mappings of the Riemann sphere $\bar{\mathbf C}$ onto itself, we introduce the notion of ${\mathcal D}$-normal family and prove that every mapping $f$ from a given Möbius invariant and ${\mathcal D}$-normal family ${\mathcal F}\subset {\mathcal D}$ is a composition of a $K$-quasiconformal automorphism of $\bar{\mathbf C}$ with the mapping, realized by a meromorphic function on $\bar{\mathbf C}$, where a constant $K$ is common for all $f\in {\mathcal F}$.

Détail

Keywords: quasiconformal mapping, mapping of bounded distortion, quasimeromorphic mapping, normal family, Möbius mapping, Möbius invariant family, Stoilov theorem, light mapping, open mapping.
Mots-clés : graph convergence