Analytic continuation for Beltrami systems, Siegel's theorem for UQR maps, and the Hilbert-Smith conjecture
The author investigates some properties of uqr-mappings. Let \(\Omega\) be a domain in \(\mathbb{R}^m\cup\{\infty\}\). A mapping \(f: \Omega\to\Omega\) is called uniformly quasiregular (or simply uqr) if \(f\) and all its iterates are quasiregular with a uniform bound on the distortion. The first result of the present paper is an analogue of Siegel's theorem on local conjugacy of rotational dynamics. Furthermore, considering the analytic continuation of some uqr-mappings the author proves the following theorem: Let \(\Omega\) be a domain in \(\mathbb{R}^m\cup\{\infty\}\) and \(f:\Omega\to\Omega\) be a uqr-mapping. Suppose that \(\partial\Omega\) has infinite cardinality. If there is a set \(V\) of homological dimension at least \(m-1\) (for example \(V\) is open) such that for \(x\in V\) we have \(f(x)= x\), then \(f(x)= x\) for all \(x\in\Omega\). From another point of view, quasiregular mappings are solutions of Beltrami systems. Applying the above theorem he shows the equivalence of solutions \(f_1\) and \(f_2\) of a Beltrami system in a domain \(\Omega\) if \(f_1\) and \(f_2\) coincide on an open subset of \(\Omega\) and if \(f_1\) is a homeomorphism. The author reduces the problems connected with these results to a central conjecture in the theory of transformation groups, namely to the Hilbert-Smith conjecture, which asserts that a locally compact acting effectively on a manifold is a Lie group. A proof of the Hilbert-Smith conjecture in the quasiconformal case was announced in another paper of the author [Electron. Res. Announc. Am. Math. Soc. 5, No. 9, 66-70 (1999; Zbl 0979.30013)].
scientific article; zbMATH DE number 1837402