The Ramanujan Journal

, Volume 13, Issue 1–3, pp 253–263 | Cite as

Möbius transformations mapping the unit disk into itself

  • Lisa Lorentzen


We consider linear fractional transformations T n which map the unit disk U into itself with the property that \(T_n(U)\subseteq T_{n-1}(U)\subseteq U\) for all n. Clearly, the closed sets \(T_n(\overline U)\) form a nested sequence of circular disks, and thus has a non-empty limit set \(T_\infty (\overline U)\). If this limit set is a single point, then {T n(w)} converges uniformly in $\overline U$ to this point. In this paper we study what happens if the limit set has a positive radius. In particular we prove that under specific conditions, the derivatives satisfy \(\sum |T_n'(w)|<\infty\)for w∈ U and {T n(w)} still converges locally uniformly in U to a constant function. Results of this type are useful in the theories of dynamical systems and continued fractions.


Linear fractional transformations Self-mappings Continued fractions Restrained sequences General convergence 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

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

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