Finiteness of the fixed point sets of automorphisms

We investigate the size of fixed point sets of automorphisms of bounded domains in ${\mathbb{C}}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be discrete. We show, under natural extension hypotheses, that discreteness forces finiteness. We also obtain a uniform bound for the number of fixed points of automorphisms in compact subgroups whose elements admit such extensions.