The fixed point set and the iterational limits of a holomorphic self-map
Let D be a convex domain in the complex Euclidean space \({\mathbb{C}}^ n,\) and \({\mathcal H}(D)\) denote the space of all holomorphic self-maps of D. Let Hol(D,\({\mathbb{C}}^ n\)) be the space of all holomorphic maps \(f: D\to {\mathbb{C}}^ n,\) and \(f^ 1=f\), \(f^ m=f\cdot f^{m-1}\). Let \(\Gamma(f)\) be the closure of the sequence \(\{f^ m\}\) in \(Hol(D,{\mathbb{C}}^ n\)) with respect to the compact open topology, \(\Gamma'(f)\) be the set of all subsequential limits of \(\{f^ m\}\), and \(Fix(f)\) be the fixed point set of f in D. The author investigates the relation between \(\Gamma(f)\) and \(Fix(f)\). Namely: Theorem 1. Let f belong to \({\mathcal H}(D)\) and \(Fix(f)\neq \emptyset\), then \(\Gamma(f)\) contains a unique holomorphic retraction \(R: D\to A\), where A is a connected complex submanifold in D, and any element g of \(\Gamma'(f)\) can be expressed as \(g=T\cdot R\) where T is a biholomorphic automorphism of A. Furthermore the dimension of A is \(d_ 1+d\) \((d_ 1=\dim Fix(f)\), \(d\geq 0).\) Theorem 2. Let f and g be commutative elements of \({\mathcal H}(D)\) with fixed points. If \(\dim Fix(f)\leq 1\) or \(\dim Fix(g)\leq 1,\) then f and g have a common fixed point. Theorem 3. Let \(f\in {\mathcal H}^*(D)={\mathcal H}(D)-\{all\) holomorphic retractions of \(D\}\). Assume that \(\Gamma(f)\) contains a holomorphic retraction \(R: D\to A\). If dim \(A\leq 1\), then f has a fixed point in D.
scientific article; zbMATH DE number 4048077
scientific article