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Commuting holomorphic maps and linear fractional models. (English) Zbl 1023.30032

Summary: Let \(f\) be a holomorphie map of the open unit disc \(\Delta\) of \(\mathbb{C}\) into itself, having no fixed points in \(\Delta\) and Wolf point \(\tau\in \partial\Delta\). In the open case in which \(f'(\tau)=1\) we study the centralizer of \(f\), i.e., the family \(G_f\) of all holomorphic maps of \(\Delta\) into itself which commute with \(f\) under composition. We prove that if the sequence of iterates \(\{f^n\}\) converges to \(\tau\) non tangentially, then \(G_f\) coincides with the set of all elements of the pseudo-iteration semigroup of \(f\) (in the sense of Cowen), whose Wolff point is \(\tau\). In the same hypotheses we give a representation of the centralizer \(G_f\) in \(\operatorname{Aut}(\Delta)\) or \(\operatorname{Aut}(\mathbb{C})\), study its main features and generalize a result due to Pranger.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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