Towards an extreme value law for the deepest cusp excursions of the unipotent flow

Abstract: The unipotent flow on the unit tangent bundle of the modular surface is a classic example of a homogeneous flow when understood through the identification with PSL_2(R)/PSL_2(Z). The ergodicity of the flow implies that almost every orbit is dense in the space and hence must eventually make excursions deeper and deeper into the cusp. We are interested in understanding the nature of these excursions. In the described setting, and more generally, Athreya and Margulis proved that the maximal excursions obey the logarithm law almost surely, meaning that their growth rate scales the logarithm of the time. In this work we focus on a more precise description of this behaviour, namely determining the probability that the deepest excursion fails to outperform the expected asymptotic behaviour by an additive amount. This question may be phrased in the language of extreme value statistics and we establish some results towards a complete extreme value law in this setting. The methods used are based on classical ideas from geometry of numbers. This is work in progress, joint with Keivan Mallahi-Karai.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar