Martin boundaries of Denjoy domains and quasiconformal mappings (Q1174387)

A domain in the plane is called a Martin domain if its complement lies on the extended real axis. The author constructs two such domains and a quasiconformal homeomorphism between them which does not have a homeomorphic extension to the Martin compactifications of the domains. The proof makes use of the Beurling-Ahlfors \(\rho\)-condition, which characterizes the boundary correspondence under quasi-conformal maps of the upper half-plane. The construction given here is simpler than an earlier similar example of Ancona. The author also quotes well-known results on compactifications of Riemann surfaces due to Sario, Nakai, and Royden.