Analytic continuation for Beltrami systems, Siegel's Theorem for UQR maps, and the Hilbert-Smith Conjecture

Abstract

A uniformly quasiregular mapping, is a mapping of the m-sphere \(\hat{Bbb R}^m ={Bbb R}^m \cup \{\infty\}\) with the property that it and all its iterates have uniformly bounded distortion. Such maps are rational with respect to some bounded measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We begin by investigating the analogue of Siegel's theorem on the local conjuga cy of rotational dynamics. We are led to consider the analytic continuation properties of solutions to the highly nonlinear first order Beltrami systems. We reduce these problems to a central and well known conjecture in the theory of transformation groups; namely the Hilbert-Smith conjecture, which roughly asserts that effective transformation groups of manifolds are Lie groups. Our affirmative solution to this problem then implies unique analytic continuation and Siegel's theorem.