Limit Sets of Semigroups of Hyperbolic Isometries

Project background and description This project is about exploring the relationship between semigroups generated by (complex) Mobius transformations, iterated function systems associated to these semigroups, and the limit sets of the semigroups. Associated to any Kleinian group is a subset of the extended complex

plane called a limit set. For a semigroup of Mobius transformations, there are both forward and backward limit sets, and their interaction can inform us about the structure of the semigroup. The forward and backward limit sets of a semigroup generated by two parabolic Mobius transformations are shown in the figure, they intersect in two

points, which correspond to the fixed points of the two parabolic maps. Semigroups and limit sets of real Mobius transformations were explored in 2. There it was found, for example, that if the forward and backward limit sets of a (real) semigroup of Mobius transformations coincide, then the semigroup is in fact a group. One challenge of this project is to

generalise that result to complex semigroups of Mobius transformations, using three-dimensional hyperbolic geometry. Another challenge is to address some of the open problems from 1. The project will begin with background reading, looking at texts such as Keen and Lakic's Hyperbolic geometry from a local viewpoint, Abate's Holomorphic dynamics on hyperbolic

Riemann surfaces, and Marden's Hyperbolic manifolds. There will also be a starter research problem to get going with alongside background reading. The project will involve programming to generate mathematical artwork. The research student will meet the principal supervisor weekly, and they will also be invited to weekly meetings of the Geometry and Dynamics Group.

Additionally, they will be able to attend the regular Dynamical Systems Seminars and other seminars