Local Quasim\"{o}bius Mappings on a Circle

V. V. Aseev ; D. G. Kuzin

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 14 (2014) no. 1, pp. 3-18

Résumé

For a family of continuous light mappings of a circle $S$ into itself it is introduced the notion ${\mathcal D}$-normality which signifies that for every graphically convergent sequence its graphical limit looks like $(Z\times S)\cup \Gamma f$, where $Z$ — zero-dimensional compact set (possibly, empty), and $\Gamma f$ is a graph of either constant mapping or continuous light mapping. It is proved that every ${\mathcal D}$-normal and Möbius invariant family of the mappings of circle $S$ into itself consist of local $\omega$-quasimöbius mappings with unified distortion function $\omega$.

Détail

Keywords: quasiconformal mapping, quasisymmetric mappings, quasimöbius mapping, local quasimöbius mapping, light mapping, graphical limit, graphical convergence, normal family of mappings, Möbius invariant families of mappings.