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Upper semicontinuity of the dimensions of automorphism groups of domains in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 125, Number 2, April 2003
- pp. 289-299
- 10.1353/ajm.2003.0011
- Article
- Additional Information
Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be the metric space of all bounded domains in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] 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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]. We also provide theorems and examples regarding the change in topological structure of these groups under small perturbation of a domain in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /].