A counterexample in formal welding concerning Hausdorff dimension (Q1107660)

Let \(\Gamma\) be a closed Jordan curve on the Riemann sphere \(\bar C,\) \(\Omega_ 1\) and \(\Omega_ 2\) the complementary components. Let \(\omega_ 1\) and \(\omega_ 2\) denote the harmonic measures on \(\Gamma\) with respect to the fixed points \(z_ 1\in \Omega_ 1\) and \(z_ 2\in \Omega_ 2\). The main theorem: For any \(1\leq d<2\) there is a quasi- circle \(\Gamma\), a \(C>0\) and points \(z_ 1\in \Omega_ 1\), \(z_ 2\in \Omega_ 2\) such that the Hausdorff dimension \(\dim (\Gamma)=d\) and for any set \(E\subset \Gamma\) \[ C^{-1}\omega_ 2(E)\leq \omega_ 1(E)\leq C\omega_ 2(E). \]