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Composition Sequences and Semigroups of Möbius Transformations

Jacques, Matthew (2016). Composition Sequences and Semigroups of Möbius Transformations. PhD thesis The Open University.

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Abstract

Motivated by the theory of Kleinian groups and by the theory of continued fractions, we study semigroups of Möbius transformations. Like Kleinian groups, semigroups have limit sets, and indeed each semigroup is equipped with two limit sets. We find that limit sets have an internal structure with features similar to the limit sets of Kleinian groups and the Julia sets of iterates of analytic functions. We introduce the notion of a semidiscrete semigroup, and find that this property is akin to the discreteness property for groups.

We study semigroups of Möbius transformations that fix the unit disc, and lay the foundations of a theory for such semigroups. We consider the composition sequences generated by such semigroups, and show that every such composition sequence converges pointwise in the open unit disc to a constant function whenever the identity element does not lie in the closure of the semigroup. We establish various results that have counterparts in the theory of Fuchsian groups. For example we show that aside from a certain exceptional family, any finitely-generated semigroup S is semidiscrete precisely when every two-generator semigroup contained in S is semidiscrete. We show that the limit sets of a nonelementary finitely-generated semidiscrete semigroup are equal (and non-trivial) precisely when the semigroup is a group. We classify two-generator semidiscrete semigroups, and give the basis for an algorithm that decides whether any two-generator semigroup is semidiscrete.

We go on to study finitely-generated semigroups of Möbius transformations that map the unit disc strictly within itself. Every composition sequence generated by such a semigroup converges pointwise in the open unit disc to a constant function. We give conditions that determine whether this convergence is uniform on the closed unit disc, and show that the cases where convergence is not uniform are very special indeed.

Item Type: Thesis (PhD)
Copyright Holders: 2016 The Author
Keywords: semigroups; transformations (mathematics); discrete groups; continued fractions
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM)
Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Item ID: 48415
Depositing User: Matthew Jacques
Date Deposited: 14 Feb 2017 14:54
Last Modified: 07 Dec 2018 10:48
URI: http://oro.open.ac.uk/id/eprint/48415
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