Upper semicontinuity of the dimensions of automorphism groups of domains in $C^n$

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 $H^n$.