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 $H^n$. 
Buma L. Fridman, Daowei Ma and Evgeny A. Poletsky