Möbius bilipschitz homogeneous arcs on the plane

Abstract

A mobius bilipschitz mapping is an η-quasimobius mapping with the linear distortion function η(t) = Kt. We show that if an open Jordan arc γ ⊂ C with distinct endpoints a and b is homogeneous with respect to the family F K of mobius bilipschitz automorphisms of the sphere C with K specified then γ has bounded turning RT(γ) in the sense of Rickman and, consequently, γ is a quasiconformal image of a rectilinear segment. The homogeneity of γ with respect to F K means that for all x, y ∈ γ {a, b} there exists f ∈ F K with f(γ) = γ and f(x) = y. In order to estimate RT(γ) from above, we introduce the condition BR(δ) of bounded rotation of γ, and then the explicit bound depends only on K and δ.