Möbius transformations mapping the unit disk into itself

Abstract

We consider linear fractional transformations T _n which map the unit disk U into itself with the property that $T_n(U)\subseteq T_{n-1}(U)\subseteq U$ for all n. Clearly, the closed sets $T_n(\overline U)$ form a nested sequence of circular disks, and thus has a non-empty limit set $T_\infty (\overline U)$. If this limit set is a single point, then {T _n(w)} converges uniformly in $\overline U$ to this point. In this paper we study what happens if the limit set has a positive radius. In particular we prove that under specific conditions, the derivatives satisfy $\sum |T_n'(w)|<\infty$for w∈ U and {T _n(w)} still converges locally uniformly in U to a constant function. Results of this type are useful in the theories of dynamical systems and continued fractions.