Departmental Colloquium
Richard Birkett & Daniel Ingebretson
UIC
Two 30 min talks
Abstract: Title for Richard Birkett: Dynamics when Zero is Divided by Zero
Abstract for Richard Birkett: What is algebraic dynamics? Generally speaking, this is the iteration of a map defined by rational functions on an algebraic space. In dimension 1, in parallel with the setting of classical complex dynamics, one typically focuses on the Fatou and Julia sets, the loci of order and chaos respectively. On an algebraic variety of any higher dimension, the dynamics of a rational map is more delicate; for instance we cannot define a Fatou-Julia theory that so nicely classifies the dynamical behaviour. Even the way to iterate is unclear: a point in motion may encounter a spot where the expression for the map appears to be zero divided by zero. What happens to the point next? Does this affect the dynamical complexity of the map? I will discuss these problems for rational maps on a complex surface, and describe how to transfer this information to a rather different dynamical system on a one-dimensional non-Archimedean space called the Berkovich projective line. By recovering a Fatou-Julia theory in the latter setting, we gain insight to the behaviour of certain rational maps on surfaces.
Title for Daniel Ingebretson: Hausdorff and packing measure of some linear numeration systems
Abstract for Daniel Ingebretson: The size of a fractal set is often measured using some form of non-integer fractal dimension. For those dimensions defined by a measure, a more subtle question is the measure of the fractal at dimension, and even for simple fractals this is largely unexplored. In this talk, we will discuss a couple of specific cases: the Hausdorff and packing measure of some restricted digit fractal sets arising in numeration systems.
Friday September 29, 2023 at 3:00 PM in 636 SEO