The limiting behavior of sequences of Möbius transformations

The Parabola Theorem on Continued Fractions

“…This terminology was introduced by Lorentzen (formerly Jacobsen) and Thron in [10]. Our definition is taken from [1] and [3,Sec. 7], and it differs from (but is equivalent to) Lorentzen's original definition.…”

Hausdorff dimension of sets of divergence arising from continued fractions

“…This terminology was introduced by Jacobsen (now Lorentzen) in [9, Definition 3.1]. Our definition is taken from [1,Theorem 3.5] and [3,Section 6], and it differs from (but is equivalent to) Lorentzen's original definition. Given another point w in H 3 , elementary hyperbolic geometry can be used to show that F n (j) converges to p if and only if F n (w) converges to p. The next basic lemma on general convergence, which we do not prove, follows from [3, Theorem 6.6].…”

“…The set of divergence is closely related to the limit set from Kleinian group theory and the Julia set from complex dynamics. Indeed, sequences such as S n associated with continued fractions share similar properties with sequences arising in Kleinian group theory and complex dynamics (see [1,3]). Whilst there is a large body of literature on the Hausdorff dimension of limit sets and Julia sets, there seems to be little known about the Hausdorff dimension of 2 IAN SHORT sets of divergence arising from continued fractions.…”

Repeated compositions of Möbius transformations

“…In our more general context, given any collection F of self maps of X, we define a composition sequence generated by F to be a sequence (F n ), where F n = f 1 f 2 · · · f n and f i ∈ F. Composition sequences generated by sets of analytic self maps of complex domains have received much attention; see, for example, [2,6,13,14]. There has been particular focus on generating sets composed of Möbius transformations that map a disc within itself -see [1,4,5,12,15,16] -partly because of applications to the theory of continued fractions. Here we give a detailed study of composition sequences generated by finite sets of Möbius transformations, which goes far beyond the existing literature in completeness and precision.…”