Local Quasimöbius Mappings on a Circle

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 $\omega$.