A rigidity theorem for special families of rational functions (Q2906167)

Let \(f\) be a nonconstant rational function, \(U\) a domain in the Riemann sphere. Let \(\mathcal H_f(U):=\{h: f^{-1}(U)\to U \text{ holomorphic and non constant}\}\). The aim of the paper under review is to find a domain \(U\) such that \(\mathcal H_f(U)=\{f\}\). For instance, \textit{L. Keen} and \textit{N. Lakic} [Contemp. Math. 432, 107--118 (2007; Zbl 1130.30030); Hyperbolic geometry from a local viewpoint. Cambridge: Cambridge University Press (2007; Zbl 1190.30001)] proved that if \(f(z)=z^2\) and \(U\) is the complement of five suitably chosen points, then \(\mathcal H_{z^2}(U)=\{z^2\}\).NEWLINENEWLINEThe author proves that if \(f\) is, up to composition with Möbius transformations, a Blaschke product with \(d\geq 3\) zeros, then there exists a domain \(U\) whose complement contains exactly \(2d+1\) points such that \(\mathcal H_f(U)=\{f\}\).NEWLINENEWLINEAlso, he gives a complete solution in case of rational functions of degree \(1\) and \(2\). For the case \(f(z)=z^n\) the set \(U\) is explicitly constructed.NEWLINENEWLINEApplications of the results are given for studying the generalized Kobayashi and Carathéodory metrics introduced by Keen and Lakic [loc. cit.]. In particular, if \(f\) is a nonconstant holomorphic map and \(\mathcal H_f(U)=\{f\}\), then the generalized Kobayashi density is defined as \(k_U^{f^{-1}(U)}(w):=\inf_{f(z)=w}\frac{\rho_{f^{-1}(U)}(z)}{|f'(z)|}\) where \(\rho_{f^{-1}}(U)\) is the hyperbolic density of \(f^{-1}(U)\). Using his constructions, mainly for \(f(z)=z^n\), the author shows some peculiarity of the generalized Kobayashi metric in comparison with the usual hyperbolic metric. For instance, he shows that \(k_X^\Omega\) can be nondegenerate on \(X\) but degenerate at certain points of \(X\), or that \(k_X^\Omega\) is in general not equivalent to the hyperbolic density, and also there may be a compact set in \(X\) having infinite area when measured in terms of the generalized Kobayashi density. Similar results are proven for the generalized Carathéodory metric.