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