Continued fractions and restrained sequences of Mobius maps

摘要

Modified approximants of a continued fraction are designed to increase the rate of convergence, and these led to the notion of restrained sequences of Mobius transformations. Here we give some analytic and geometric characterizations of restrained sequences and related topics. We also give an expository account of the use of geometry, including hyperbolic geometry, in discussing restrained sequences and continued fractions.