Parts in $H^{\infty }$ with homeomorphic analytic maps

Abstract:

Hoffman characterized the parts of ${H^\infty }$ as either singleton points or analytic discs. He showed that a part belongs to the latter category if and only if it is hit by the closure of an interpolating sequence and that there are cases where a corresponding analytic map is a homeomorphism and cases where it is not. We show that there is no class, $\mathcal {C}$, of subsets of the open unit disc such that an analytic map of a part $P$ is a homeomorphism if and only if $P$ is hit by the closure of some set in $\mathcal {C}$.