On holomorphic maps which commute with hyperbolic automorphisms
Suppose \(\gamma\) is a holomorphic automorphism of the unit disk \(\Delta\) in \({\mathbb C}\) and suppose \(f\in \text{Hol}(\Delta,\Delta)\) with \(f \circ \gamma = \gamma \circ f\). \textit{M. H. Heins} [Duke Math. J. 8, 312-316 (1941; Zbl 0025.17301)] showed that either \(f = \text{id}\) or \(f\) is also a hyperbolic automorphism of \(\Delta\) with the same fixed points as \(\gamma\). This paper provides some insight into a similar problem for \(\Delta_n\), the unit ball in \({\mathbb C}^n\), when \(n>1\). The first observation is that the two fixed points \(p_1\) and \(p_2\) of \(\gamma\) are ``fixed points'' of \(f\), in the sense of \(K\)-limits. By conjugating in \(\text{Aut}(\Delta_n)\) we may suppose that these fixed points are \(e_1\) and \(-e_1\). Under the assumption of a ``regularity condition'', i.e., the existence of a certain \(K\)-limit at \(e_1\), the authors show that \(f_1\) (the first component function of \(f\)) does not depend on \(z_2, \ldots, z_n\). An explicit formula for \(f_1\) is given.
scientific article; zbMATH DE number 1334585
scientific article