Multivalued quasimöbius mappings from circle to circle

We prove that if a multivalued mapping $F$ of circle to circle has the $\eta$-BAD property (bounded distortion of generalized angles with control function $\eta$) then there exist a positive integer $N$ and a quasimöbius homeomorphism $\varphi$ of a circle into itself such that the left inverse mapping to $F$ is of the form $(\varphi(z))^N$. Moreover, $\varphi$ is a locally $\omega$-quasimöbius mapping with $\omega$ depending only on $\eta$ and $N$.