Repeated compositions of Moebius transformations

TL;DR

This paper studies dynamical systems generated by finite sets of Moebius transformations on the unit disc, classifying sequences by their limit types, and analyzing their convergence and Hausdorff dimension.

Contribution

It provides necessary and sufficient conditions for sequences to be of limit-disc type and calculates their Hausdorff dimension in key cases.

Findings

Sequences of limit-disc type are rare and occur under specific conditions.

The paper establishes criteria for convergence of these dynamical systems.

Hausdorff dimension of limit-disc sequences is computed in significant cases.

Abstract

This paper considers a class of dynamical systems generated by finite sets of Moebius transformations acting on the unit disc. Compositions of such Moebius 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 particular 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.