Properties of Fixed Point Sets and a Characterization of the Ball in ${\mathbb{C}}^n$

We study the fixed point sets of holomorphic self-maps of a bounded domain in ${\mathbb{C}}^n$. Specifically we investigate the least number of fixed points in general position in the domain that forces any automorphism (or endomorphism) to be the identity. We have discovered that in terms of this number one can give the necessary and sufficient condition for the domain to be biholomorphic to the unit ball. Other theorems and examples generalize and complete previous results in this area, especially the recent work of Jean-Pierre Vigu'{e}.