Repeated compositions of Möbius transformations

Jacques, Matthew and Short, Ian (2020). Repeated compositions of Möbius transformations. Ergodic Theory and Dynamical Systems, 40(2) pp. 437–452.

DOI: https://doi.org/10.1017/etds.2018.43

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.

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