A Survey of Dynamical Billiards

Sammanfattning: In this report we will present the basic concepts and results of the theory of dynamical billiards which idealizes the concept of a volumeless ball re　ecting against the inside of a billiard table without friction.This motion will continue indenitely and it is of interest to study its behaviour. We will show that the study of a billiard system can be reduced to the study of an associated map called the billiard map dened on a cylindrical phase space. Using this formalism the specic systems where the billiard table is given by a circle, right iscoceles triangle and ellipse will be studied in some detail along with the existence of peridic points through Birkho's famous theorem and some more novel results such as an instance of Benford's law regarding the distribution of rst digits in real-life data. We will also dene the concept of a caustic and investigate their existence and non-existence which will lead us to the concept of circle homeomorphisms and will also provide the opportunity to illustrate the systems with some simulations and yield some more informal and practical insight into the behaviour of these systems.