Abstract:
A möbius bilipschitz mapping is an $\eta$-quasimöbius mapping with the linear distortion function $\eta(t)=Kt$. We show that if an open Jordan arc $\gamma\subset\overline{\mathbb C}$ with distinct endpoints $a$ and $b$ is homogeneous with respect to the family $\mathscr F_K$ of möbius bilipschitz automorphisms of the sphere $\overline{\mathbb C}$ with $K$ specified then $\gamma$ has bounded turning $RT(\gamma)$ in the sense of Rickman and, consequently, $\gamma$ is a quasiconformal image of a rectilinear segment. The homogeneity of $\gamma$ with respect to $\mathscr F_K$ means that for all $x,y\in\gamma\setminus\{a,b\}$ there exists $f\in\mathscr F_K$ with $f(\gamma)=\gamma$ and $f(x)=y$. In order to estimate $RT(\gamma)$ from above, we introduce the condition $BR(\delta)$ of bounded rotation of $\gamma$, and then the explicit bound depends only on $K$ and $\delta$.
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