  Let $H^n$ be the metric space of all bounded domains in $C^n$ with the metric
equal to the Hausdorff distance between boundaries of domains. We prove that
the dimension of the group of automorphisms of domains is an upper
semicontinuous function on $\H^n$. We also provide theorems and examples
regarding the change in topological structure of these groups under small
perturbation of a domain in $H^n$.
