Limit functions for convergence groups and uniformly quasiregular maps

We investigate the limit functions of iterates of a function belonging to a convergence group or of a uniformly quasiregular mapping. We show that it is not possible for a subsequence of iterates to tend to a non-constant limit function, and for another subsequence of iterates to tend to a constant limit function. It follows that the closure of the stabiliser of a Siegel domain for a uniformly quasiregular mapping is a compact abelian Lie group, which we further conjecture to be infinite. This result concerning possible limits of convergent subsequences of iterates for holomorphic rational functions on the Riemann sphere is known, and the only known method of proof involves universal covering surfaces and Möbius groups. Hence, our method yields a new and perhaps more elementary proof also in that case.