Consider a conformal homeomorphism $f \colon \Omega \to \Omega'$ between domains $\Omega, \Omega'$ in the plane. Let $E = \mathbb{R}^2 \setminus \Omega$ and $E' = \mathbb{R}^2 \setminus \Omega'$. My understanding is that you can have a situation where $E$ is totally disconnected and yet $E'$ is not. If we think of the map $f$ as extending to a map between the connected components of $E$ and the connected components of $E'$, then $f$ would take a point $x \in E$ to a non-trivial continuum in $E'$. 


Is there a standard, more-or-less explicit example of such a set $E$ and map $f$? 



Conformal mappings for domains whose complement is totally disconnected
The only example I know of a conformal map that "stretches" a point boundary component to a nondegenerate continuum was constructed by Gehring and Martio in *Quasiextremal distance domains and extension of quasiconformal mappings*, J. Analyse Math. 45 (1985), 181–206. See Theorem 4.1.


The authors use earlier results of Ahlfors and Beurling to construct a Cantor set $E$ of positive area whose complement is mapped conformally to a domain whose boundary components are single points and the unit circle. The construction is based on the following classical result relating the existence of such a map with extremal length considerations:


**Theorem**
Let $\Omega \subset \mathbb{C}$ be a domain and let $z\_0 \in \partial \Omega$ be a point boundary component. The following are equivalent :


* There is a conformal map on $\Omega$ that sends $z\_0$ to non-degenerate boundary component of $f(\Omega)$;
* There exists $r>0$ such that $\operatorname{mod}(\Gamma) < \infty$, where $\Gamma$ is the family of all closed curves in $\Omega \cap B(z\_0,r)$ having nonzero winding number around $z\_0$.


The idea is to start with a single point, say $0$, and surround it by sufficiently many point boundary components in order to block the curves winding around $0$ so that the modulus is finite. This implies that $0$ is mapped by a nondegenerate continuum under some conformal map $f$. 


Lastly, let me mention that your question, more specifically the existence of such a set $E$ which is small (at least zero area), is related to a very difficult conjecture of He and Schramm on the rigidity of circle domains in Koebe's uniformization conjecture. See Section 9 of my paper with Ntalampekos


D. Ntalampekos and M. Younsi, *Rigidity theorems for circle domains*, Invent. Math. 2019


