Propriedades ergódicas do bilhar no estádio elíptico

Abstract

In the development of the theory of billiards, the search for ergodic billiards was a subject of intense study, first with dispersive billiards and later with the focusing-type ones. The first example of an ergodic billiard of focusing type was the estadium, introduced by Bunimovich in [5]. Soon after, a generalization of this stadium was proposed, called elliptical stadium. The elliptical stadium is a convex region of the plane whose boundary consists of two semielipses with semi-axes of length 1 and a, a > 1 joined by two straight lines of length 2h, which are parallel to the major axis of the semi-ellipses. Donnay, in [15], proved some properties of the elliptical stadium billiard and launched a challenge to find optimal estimates for the lengths of a and h to ensure ergodicity of billiards. Canale, Del Magno, Markarian, Oliffson and Pinto did work on this challenge and found good estimates . The objective of this dissertation is to prove that when a and h satisfy ,FÓRMULA DISPONÍVEL NO TEXTO COMPLETO, the billiard on the elliptical stadium is ergodic.