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Repeated compositions of Möbius transformations

Jacques, Matthew and Short, Ian (2019). Repeated compositions of Möbius transformations. Ergodic Theory and Dynamical Systems (In Press).

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Abstract

We consider a class of dynamical systems generated by finite sets of Möbius transformations acting on the unit disc. Compositions of such Möbius transformations give rise to sequences of transformations that are used in the theory of continued fractions. In that theory, the distinction between sequences of limit-point type and sequences of limit-disc type is of central importance. We prove that sequences of limit-disc type only arise in exceptional circumstances, and we give necessary and sufficient conditions for a sequence to be of limit-disc type. We also calculate the Hausdorff dimension of the set of sequences of limit-disc type in some significant cases. Finally, we obtain strong and complete results on the convergence of these dynamical systems.

Item Type: Journal Item
Copyright Holders: 2018 Cambridge University Press
ISSN: 1469-4417
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 55135
Depositing User: Ian Short
Date Deposited: 21 May 2018 08:13
Last Modified: 27 Jun 2018 08:33
URI: http://oro.open.ac.uk/id/eprint/55135
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