Dynamics on weakly pseudoconvex domains
In the given paper the iterative behaviour of holomorphic selfmappings \(f\in H(\Omega, \Omega)\) on suitable domains \(\Omega\in \mathbb{C}^2\) is studied if \((f^n )_{n\in \mathbb{N}}\) (the sequence of iterates) is normal on \(\Omega\). In \S 2 the authors give the following characterization on weakly pseudo-convex domains if \(f\) has some nonwandering point \(p\in \Omega\): Either \(p\) is an attracting fixpoint of \(f\). Or \(f\) is an isomorphism on some submanifold \(S\) of dimension 1 and some subsequence of \((f^n)\) converges to the identity on \(S\). Or \(f\) is an automorphism on \(\Omega\) and some subsequence of \((f^n)\) converges to the identity on \(\Omega\). This is in some sense a generalization of some result of \textit{P. Hriljac} [Nonlinear Anal. Theory Methods Appl. 19, No. 8, 717-730 (1992; Zbl 0771.32014)]. \S 3 handles the natural question what happens without the assumption on the existence of some nonwandering point. The following theorem of Denjoy-Wolff-type is shown under the additional assumption that the boundary of \(\Omega\) of real-analytic: Either \((f^n)\) converges to some point in \(\overline {\Omega}\) uniformly on compact subsets of \(\Omega\). Or there exists some uniquely defined holomorphic retraction \(R\) in the set of limit functions of \((f^n)\) such that for any limit function \(g\) there exists an automorphism \(T\) on the submanifold \(R(\Omega)\) with \(g= T\circ R\).
scientific article; zbMATH DE number 832248