Teorema da curva invariante de Birkhoff e bilhares não-elásticos

Abstract

One of the objectives of this paper was to understand Birkhoff's Invariant Curve Theorem which was first demonstrated by Birkhoff himself and has as an important consequence that every invariant rotational curve projects injectively over $ S ^ 1 $. In addition, we will present billiards, denoted non-elastic billiards, which have a modified law of reflection, corresponding to a contraction in the vertical fibers of an invariant rotational curve. These consist of simple examples of dynamic systems with limit set having dominated decomposition. We will prove that under some assumptions of differentiability and some limits in contraction, there is a compact range in phase space, where the application of non-elastic billiard map is a $C^2$ diffeomorphism.