Attractors in quasiregular semigroups

Abstract

Suppose that $f$ generates a $K$-quasimeromorphic semigroup in a domain $D$ of $overline{{R}^n}$, where $n ge 2$. Suppose that $U$ is a topological ball with $overline{f(U)} subset U$ and $overline{U} subset D$, and that $f|U$ is a homeomorphism. We prove that then $U$ contains a unique fixed point $w$ of $f$ (so that $f(w) = w)$, and there is a topological ball neighbourhood $V$ of $w$ with $overline{V} subset U$ and a quasiconformal homeomorphism $g$ of $overline{{R}^n}$ onto itself with $g(w)=0$ such that $(g circ f circ g^{-1})(x)=z/2$ for all $xin g(V)$. this allows us to classify the attracting and repelling fixed points of elements of uniformly quasimeromorphic semigroups such that the element is quasiconformally conjugate to a dilation in a neighbourhood of such a point.