טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKassis Ameer
SubjectEquidistribution Through Thickening and Arithmetic Extension
DepartmentDepartment of Mathematics
Supervisor Professor Uri Shapira
Full Thesis textFull thesis text - English Version


Abstract

The main object of this thesis is to display two applications of the thickening trick around compact orbits in homogeneous spaces. In particular, we use it in order to prove two results concerning equidistribution of some families of the horocycle and geodesic orbits in the space of the unimodular lattices in the two dimensional Euclidean space. We start by introducing lattices in the n-dimensional real Euclidean space Rn, mainly as a motivation for the more general case, which is lattices in locally compact groups. A well known fact is that locally compact groups carry invariant measures with respect to translations. This establishes the existence of invariant measures on their homogeneous spaces, with respect to the group actions. After introducing the building blocks, we characterize and study periodic horocycle lattices, and prove that they are equidistributed. Along the way, we prove a special case of the Howe-Moore theorem, and present the notion of tubes which is the basic object of the thickening trick. Using tools from dynamical analysis on p-arithmetic homogeneous spaces, we establish a similar result regarding geodesics.