Quasiregular semigroups

We study the conformal mappings of a given measurable conformal structure on the Riemann sphere. We construct an example of a quasiregular self mapping of the n-sphere whose iterates have uniformly bounded dilatation with nonempty branch set. We describe the Fatou and Julia sets of this function and discuss the associated invariant measurable conformal structures as well as some simple dynamical properties. We thereby deduce that conformal mappings between the same measurable structure need not be locally homeomorphic.