Date: 22 September 2022
Speaker: Simon Baker
Title: Overlapping iterated function systems from the perspective of Metric Number Theory
Abstract: Khintchine’s theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the divergence of naturally occurring volume sums. Importantly this result provides a quantitative description of how the rationals are distributed within the reals. In this talk I will discuss some recent work where I prove that a similar Khintchine like phenomenon occurs typically within many families of overlapping iterated function systems. Families of iterated function systems these results apply to include those arising from Bernoulli convolutions, the 0,1,3 problem, and affine contractions with varying translation parameters.
Time permitting I also will discuss a particular family of iterated function systems for which we can be more precise. Our analysis of this family shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.
Date: 6 October 2022
Speaker: Juno Seong
Title: An avoidance principle and Margulis functions for expanding translates of unipotent orbits
Abstract: Avoidance principles — quantifying how much time trajectories avoid certain subsets of the ambient space — have been fruitful in the study of dynamical systems. We prove an avoidance principle for expanding translates of unipotent orbits for some semisimple homogeneous spaces. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed orbits of bounded volume. The proof of our results relies on the construction of a Margulis function and the theory of finite dimensional representations of semisimple Lie groups. This is joint work with Anthony Sanchez.
Date: 17 November 2022
Speaker: Nicolas de Saxce
Title: Rational approximations to linear subspaces
Abstract: Using diagonal orbits on the space of lattices, we revisit some old questions of Schmidt concerning diophantine approximation on Grassmann varieties, and in particular, we prove a version of Dirichlet’s principle in that setting.