For a smoothly bounded pseudoconvex domain $D\subset{\mathbb
C}^n$ of finite type with non-compact holomorphic automorphism group
$\hbox{Aut}(D)$, we show that the set $S(D)$ of all boundary
accumulation points for $\hbox{Aut}(D)$ 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.