Statistical Properties of Dynamical Systems

: From Statistical Mechanics to Hyperbolic Geometry

Abstract

Broadly, this thesis treats the statistical properties of dynamical systems in two different contexts. That is, we characterise asymptotic behaviour, independence, and randomness in two distinct settings.

First we consider two models for diffusion of gasses: the random Lorentz gas and the random wind-tree model. Understanding how typical particles diffuse outwards is one of the central aims of the field. In both these contexts our main results state that (in a particular scaling limit), when considered over long times, the typical particle trajectory converges in distribution to a Brownian motion. We use novel coupling methods to approximate these trajectories by Markovian walks and thus prove these invariance principles.

For the second half, we consider a general discrete hyperbolic subgroup. Therefore these groups may be 'thin'. Then we consider the orbit of a point in hyperbolic half-space by this group. The main results concern characterising the limiting local statistics of these orbits in a number of different contexts. We extend methods from homogeneous dynamics to the thin group setting and use Patterson-Sullivan theory and equidistribution of expanding horospheres to characterise the limiting behaviour of these group orbits. This work has applications to sphere packings, Diophantine approximation, continued fractions, and more.

Date of Award	24 Mar 2020
Original language	English
Awarding Institution	University of Bristol
Supervisor	Jens Marklof (Supervisor) & Francesco Mezzadri (Supervisor)