On determining sets for holomorphic automorphisms (Q2477862)

A subset \(K\) of the closure of a bounded domain \(D\subset\subset \mathbb C^n\) with \(C^1\) boundary is said to be a determining subset if whenever \(g\) is an automorphism of \(D\), continuous up to the closure, and \(g(z)=z\) for all \(z\in K\), then \(g= \text{Id}\). For instance, in the unit disc of \(\mathbb C\), two points of the closed disc, one of which in the open disc, form a determining subset. In this paper the authors study determining subsets of bounded domains in higher dimension. The main result they prove is that, except for a set of zero Hausdorff-Lebesgue measure in \(D\times (\partial D)^n\), every choice of \(n+1\) points form a determining subset of \(\overline{D}\). They conclude the paper with some corollary of this result and examples.