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Abstract: Inspired by questions of convergence in continued fraction theory, Erd\H{o}s,Piranian and Thron studied the possible sets of divergence for arbitrarysequences of M\-obius maps acting on the Riemann sphere, $S^2$. By identifying$S^2$ with the boundary of three-dimensional hyperbolic space, $H^3$, we showthat these sets of divergence are precisely the sets that arise as conicallimit sets of subsets of $H^3$. Using hyperbolic geometry, we give simplegeometric proofs of the theorems of Erd\H{o}s, Piranian and Thron thatgeneralise to arbitrary dimensions. New results are also obtained about theclass of conical limit sets, for example, that it is closed under locallyquasisymmetric homeomorphisms. Applications are given to continued fractions.



Autor: Edward Crane, Ian Short

Fuente: https://arxiv.org/







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