.A mathematical billiard is the mechanical system of a point particle traveling in a planar domain subject only to elastic collisions with the boundary of the domain (assumed to be infinitely massive). Systems of this kind have wide application in the physical sciences. I will start by giving a light but mathematically rigorous description of a billiard system, trying to show what features of this description lead to the observed properties of the dynamics (e.g., regular, chaotic, etc.). Ergodic theory is the branch of mathematics that studies the properties of a “dynamical system” from a stochastic/probabilistic point of view. It is the main and perhaps only mathematical tool to study systems, such as chaotic billiards, that admit no exact prediction. I will introduce the most basic notions of ergodic theory, together with some elementary examples of dynamical systems, with a view towards the paradigm of the chaotic billiard.
7 Febbraio 2019
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