On Belinskii conformality in countable sets of points (Q2723504)

The authors study quasiconformal mappings defined in the neighborhood of a point in the context of Belinskii conformality, such a mapping is called differentiable in the sense of Belinskii if \((w= f(z))\) NEWLINE\[NEWLINE\Delta w= A(\rho)(\Delta z+ \mu_0\Delta\overline z)+ o(\rho),NEWLINE\]NEWLINE where \(A(\rho)\) depends on \(\rho= |\Delta z+\mu_0\Delta\overline z|\) with \(\lim_{\rho\to 0} {A(t\rho)\over A(\rho)}= 1\) for each field \(t>0\). \(f\) is said to be conformal in the sense of Belinskii of \(\mu_0= 0\).NEWLINENEWLINENEWLINEThe authors remark that conformality in the sense of Belinskii is equivalent (at the point \(0\), \(f(0)= 0\)) to \(\lim_{z\to 0} {b(z\zeta)\over f(z)}= \zeta\) for each complex \(\zeta\) or to \(\lim_{t\to 0} {f(zt)\over f(z)}= t\) for each fixed \(t>0\). Their principal result is as follows.NEWLINENEWLINENEWLINELet \(K(z)\) be an arbitrary measurable function from the complex plane to the interval \([1,Q]\). Then there exists a quasiconformal mapping of the plane onto itself with local dilation \(f(z)= K(z)\) a.e. such that the mapping is conformal in the sense of Belinskii at a prescribed countable set of points.NEWLINENEWLINENEWLINEThe authors refer to the property of a homeomorphism defined in a neighborhood of a point being an asymptotic rotation on circles introduced by \textit{M. Brakalova} and the reviewer [Kodai Math. J. 17, 201--213 (1994; Zbl 0811.30015)]. To prove this condition at a point is a stronger result than to prove Belinskii conformality.