Quasiregular analogues of critically finite rational functions with parabolic orbifold
Let \(\Gamma\) be a discrete group of isometries of \(\mathbb{R}^n\) and let \(h:\mathbb{R}^n\to \overline{\mathbb{R}}^n\) a quasiregular automorphic mapping with respect to \(\Gamma\) [\textit{O. Martio} and \textit{U. Srebro}, J. Anal. Math. 28, 20-40 (1975; Zbl 0317.30025)]. There exists a uniformly quasiregular mapping \(f\) of \(\overline{\mathbb{R}}^n\) satisfying Schröder's equation \(f\circ h=h\circ A\) where a given similarity \(A\) is such that \(A\circ \Gamma\circ A^{-1}\subset \Gamma\); \(f\) is called a mapping of Lattès-type. The non-injective Lattès-type mapping \(f\) is characterized by the property that it has an \(f\)-invariant conformal structure which is flat at the repelling fixed point of some iterate \(f^k\) of \(f\). The paper also contains interesting examples of Lattés-type mappings.
scientific article; zbMATH DE number 1221951
scientific article