  For a smoothly bounded pseudoconvex domain $D\subset{\Bbb C}^n$ of finite
type with non-compact holomorphic automorphism group $\text{Aut}(D)$, we show
that the set $S(D)$ of all boundary accumulation points for $\text{Aut}(D)$ is
a compact subset of $\partial D$ and, if $S(D)$ contains at least three points,
it is connected and thus has the power of the continuum. We also discuss how
$S(D)$ relates to other invariant subsets of $\partial D$ and show in
particular that $S(D)$ is always a subset of the \v{S}ilov boundary.
