Convergence properties of sequences of linear fractional transformations.

Normal Families of Möbius Maps

“…It is known that if a sequence of Möbius maps converges at three distinct points to three distinct values, then it converges uniformly on C ∞ to a Möbius map [18]. We sketch a proof of this fact that is reminiscent of standard proofs of Vitali's theorem using a normal families argument.…”

Hausdorff dimension of sets of divergence arising from continued fractions

“…The set of divergence of S n is the set of points z in C ∞ for which the sequence S n (z) diverges. Sets of divergence for general sequences of Möbius transformations have been studied in [5,6,14]. The set of divergence is closely related to the limit set from Kleinian group theory and the Julia set from complex dynamics.…”

Conical limit sets and continued fractions

“…In [11] George Piranian and Wolfgang Thron defined a class of subsets of the Riemann sphere C ∞ , which they called sets of divergence. A set A ⊆ C ∞ is a set of divergence if and only if there exists a sequence (G n ) of Möbius transformations such that the sequence (G n (z)) diverges for each z ∈ A and converges for each z ∈ A. Paul Erdős and Piranian continued the study of these sets in [9], giving a geometric characterisation of the countable sets of divergence.…”

“…Proposition 1.1 enables us to give purely geometric proofs of the known results about sets of divergence, and also some new results about these sets, without reference to explicit sequences of Möbius maps. Most of the proofs of [9,11] involve explicit sequences of Möbius transformations, hence these proofs are valid only in two-dimensions. In contrast, our geometric proofs are valid in all dimensions.…”