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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Infinite compositions of Möbius transformations


Author: John Gill
Journal: Trans. Amer. Math. Soc. 176 (1973), 479-487
MSC: Primary 30A22; Secondary 40A15
MathSciNet review: 0316690
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Abstract: A sequence of Möbius transformations $ \{ {t_n}\} _{n = 1}^\infty $, which converges to a parabolic or elliptic transformation t, may be employed to generate a second sequence $ \{ {T_n}\} _{n = 1}^\infty $ by setting $ {T_n} = {t_1} \circ \cdots \circ {t_n}$. The convergence behavior of $ \{ {T_n}\} $ is investigated and the ensuing results are shown to apply to continued fractions which are periodic in the limit.


References [Enhancements On Off] (What's this?)

  • [1] Michael Mandell and Arne Magnus, On convergence of sequences of linear fractional transformations, Math. Z. 115 (1970), 11–17. MR 0258976
  • [2] L. R. Ford, Automorphic functions, McGraw-Hill, New York, 1929.
  • [3] T. J. I'A.Bromwich, An introduction to the theory of infinite series, Macmillan, London, 1947.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0316690-6
Keywords: Hyperbolic, loxodromic, elliptic Möbius transformations, parabolic Möbius transformations, continued fractions periodic in the limit
Article copyright: © Copyright 1973 American Mathematical Society