Quasiregular semigroups with examples (Q1737077)

Quasiregular semigroups are generalizations of rational semigroups [\textit{A. Hinkkanen} and \textit{G. J. Martin}, Proc. Lond. Math. Soc. (3) 73, No. 2, 358--384 (1996; Zbl 0859.30026)]. A quasiregular (quasimeromorphic) mapping \(f: S^n \rightarrow S^n\), \(S^n = \overline{\mathbb{R}}^n\), offers a substitute for a rational function to higher-dimensional Euclidean spaces. Such a non-constant mapping \(f\) has its maximal dilatation \(K(f) > 1\) provided that the branch set of \(f\) is \(\neq \emptyset\) and \(n \geq 3\). This makes it difficult to construct uniformly quasiregular mappings \(f\), i.e., \(K(f^m) \leq C\) for all \(m\) where \(f^m\) denotes the \(m\)-fold iteration of \(f\). If \(f\) is a uniformly quasiregular mapping, then \(G = \{ f^m: m \in \mathbb{N} \}\) is a quasiregular semigroup and, more generally, \(G\) is a \(K\)-quasiregular semigroup in \(S^n\), \(n \geq 2\), if each \(g \in G\) is \(K\)-quasiregular, non-injective and closed under composition. It is shown that if \(G\) is a \(K\)-quasiregular semigroup generated by \(\{g_i : i \in I \}\) and \(\mathrm{Lip}_{1/K}(g_i) \leq C\) for all \(i \in I\), then the Julia set \(J(G)\) is uniformly perfect. Here \(\mathrm{Lip}_{1/K}\) refers to the Hölder condition of order \(1/K\) in terms of the chordal metric in \(S^n\) and \(J(G)\) is the set of points \(x \in S^n\) which do not have a neighborhood \(U\) such that \(G\) restricted to \(U\) is a normal family. The proof uses a variant of Rickman's Picard theorem in \(\mathbb{R}^n\), [\textit{S. Rickman}, Quasiregular mappings. Berlin: Springer-Verlag (1993; Zbl 0816.30017)]. Thus for a finitely generated semigroup \(G\), the Julia set is always uniformly perfect. Several examples of uniformly quasiregular mappings and quasiregular semigroups \(G\) are provided. For example, there is \(G\), \(n \geq 2\), such that \(J(G)\) is a closed annulus \(\{ 1 \leq |x| \leq a \}\), \(a > 1\), and another \(G\) so that \(J(G)\) is topologically a product of a Cantor set and an \((n-1)\)-sphere and the Hausdorff dimension of \(J(G)\) is at least \(n-\varepsilon\).