  Let $G$ be a finite union of disjoint and bounded Jordan domains in the
complex plane, let $\mathcal{K}$ be a compact subset of $G$ and consider the
set $G^\star$ obtained from $G$ by removing $\mathcal{K}$; i.e.,
$G^\star:=G\setminus \mathcal{K}$. We refer to $G$ as an archipelago and
$G^\star$ as an archipelago with lakes. Denote by $\{p_n(G,z)\}_{n=0}^\infty$
and $\{p_n(G^\star,z)\}_{n=0}^\infty$, the sequences of the Bergman polynomials
associated with $G$ and $G^\star$, respectively; that is, the orthonormal
polynomials with respect to the area measure on $G$ and $G^\star$. The purpose
of the paper is to show that $p_n(G,z)$ and $p_n(G^\star,z)$ have comparable
asymptotic properties, thereby demonstrating that the asymptotic properties of
the Bergman polynomials for $G^\star$ are determined by the boundary of $G$. As
a consequence we can analyze certain asymptotic properties of $p_n(G^\star,z)$
by using the corresponding results for $p_n(G,z)$, which were obtained in a
recent work by B. Gustafsson, M. Putinar, and two of the present authors. The
results lead to a reconstruction algorithm for recovering the shape of an
archipelago with lakes from a partial set of its complex moments.
