Iteration theory in hyperbolic domains (Q1128158)

Some of the deep results proved in 1981, 1984 by \textit{C. Cowen} [Trans. Am. Math. Soc. 265, 69-95 (1981; Zbl 0476.30017); ibid. 283, 685-695 (1984; Zbl 0542.30030)] about iteration and fixed-points of holomorphic self-maps of the disk \( \Delta\) are extended to hyperbolic domains of regular type. These are noncompact domains \(D\) lying on a compact Riemann surface \(\widehat X\) such that (i) every connected component of \(\partial D\) is either a Jordan curve or a single point and (ii) every such component can be separated from all the rest by an open subset of \(\widehat X\). Effectively this class of domains contains all finitely multiply connected hyperbolic domains. Naturally heavy use is made of the universal covering of \(D\) by \(\Delta\). A 1970 result of \textit{W. Pranger} [Aequationes Math. 4, 201-204 (1970; Zbl 0197.35004] is also generalized to \(D\). This is a characterization of sets \(\{g'(z_0): g\in \text{Hol} (\Delta)\), \(g\circ f= f\circ g\}\) arising from some locally univalent \(f\in\text{Hol} (\Delta, \Delta)\) which fixes \(z_0\). A representative Cowen result that is generalized concerns the existence of a fundamental set for each fixed-point-free \(f\in\text{Hol} (\Delta, \Delta)\). This is an open, connected, simply-connected \(V\subset \Delta\) which \(f\) maps injectively into itself and which devours the tails of \(f\)-orbits of all compact \(K\subset \Delta\): for all sufficiently large \(n\), the \(n\)th iterate of \(f\) maps \(K\) into \(V\).