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 windtree 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 halfspace 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 PattersonSullivan 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 

Supervisor  Jens Marklof (Supervisor) & Francesco Mezzadri (Supervisor) 
Keywords
 Dynamical systems
 Probability