Computing boundary extensions of conformal maps part 2

TL;DR

This paper demonstrates that a conformal map can be computable with a computable extension to the boundary even if the boundary's local connectivity isn't effectively computable, by encoding a c.e. set into the boundary's local connectivity.

Contribution

It constructs a conformal map with a computable extension despite non-effective local connectivity of the boundary, encoding a c.e. set into the boundary's local connectivity.

Findings

Existence of a computable conformal map with computable boundary extension

Boundary of the domain can be non-effectively locally connected

Encoding of a c.e. set into boundary local connectivity

Abstract

It is shown that there is a computable conformal map of the unit disk onto a domain $D$ that has a computable extension to the closure of the unit disk even though the boundary of $D$ is not effectively locally connected. The proof encodes an arbitrary \emph{c.e.} set into the local connectivity of the boundary of $D$.