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Constructive Approximation

, Volume 38, Issue 2, pp 171–191 | Cite as

Limiting Behavior of Random Continued Fractions

  • Lisa Lorentzen
Article

Abstract

Let K(a n /b n ) be a continued fraction with elements (a n ,b n ) picked randomly and independently from \((\mathbb{C}\setminus\{0\})\times\mathbb{C}\) according to some probability distribution μ. We find sufficient conditions on μ for K(a n /b n ) to converge with probability 1 or to be restrained with probability 1. More generally, we also consider μ-random sequences {τ n } of independent Möbius transformations and find sufficient conditions for \(\{\tau_{1}\circ\tau_{2}\circ\cdots\circ\tau_{n}\}_{n=1}^{\infty}\) to converge or be restrained with probability 1. The analysis is based on an important paper by Furstenberg.

Keywords

Iterated function systems Random Möbius transformations Linear fractional transformations General convergence Restrained sequences Random composition sequences Random continued fractions Furstenberg’s theorem 

Mathematics Subject Classification

40A15 37A30 15A51 20H05 60F99 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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