Limit Functions for Convergence Groups and Uniformly Quasiregular Maps

We investigate the limit functions of iterates of a functionbelonging to a convergence group or of a uniformly quasiregularmapping. We show that it is not possible for a subsequence ofiterates to tend to a non-constant limit function, and for anothersubsequence of iterates to tend to a constant limit function.It follows that the closure of the stabiliser of a Siegel domainfor a uniformly quasiregular mapping is a compact abelian Liegroup, which we further conjecture to be infinite. This resultconcerning possible limits of convergent subsequences of iteratesfor holomorphic rational functions on the Riemann sphere isknown, and the only known method of proof involves universalcovering surfaces and Möbius groups. Hence, our methodyields a new and perhaps more elementary proof also in thatcase.