Conical limit sets and continued fractions

Crane, Edward; Short, Ian

doi:10.1090/S1088-4173-07-00169-5

Conical limit sets and continued fractions
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by Edward Crane and Ian Short

Conform. Geom. Dyn. 11 (2007), 224-249

DOI: https://doi.org/10.1090/S1088-4173-07-00169-5

Published electronically: October 31, 2007

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Abstract:

Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with the boundary of three-dimensional hyperbolic space, $H^3$, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of $H^3$. Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets; for example, it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.

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