Hausdorff dimension of sets of divergence arising from continued fractions

Short, Ian (2012). Hausdorff dimension of sets of divergence arising from continued fractions. Proceedings of the American Mathematical Society, 140(4) pp. 1371–1385.

DOI: https://doi.org/10.1090/S0002-9939-2011-11032-3

Abstract

A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We determine the Hausdorff dimensions of sets of divergence for sequences of Möbius transformations corresponding to certain important classes of continued fractions.

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