O bilhar no limão assimétrico: uma nova percepção sobre hiperbolicidade

Abstract

Billiards are dynamical systems that model the motion of a particle reflecting off the boundaries of a region, obeying the law of reflection. The study of the chaotic behavior of these systems dates back to the 1970s, through the seminal article of Yakov Sinai. The geometric character of these systems favors the analysis of transitions between regular and chaotic behaviors, with emphasis on the study of hyperbolicity. Among the classic examples, the class of Bunimovich Flowers stands out, focusing billiards whose components satisfy the Defocusing Mechanism proposed by Leonid Bunimovich, that is, their components are sufficiently distant from each other, resulting in the hyperbolicity of the system. In this work, we will demonstrate the hyperbolicity of the Asymmetric Lemon Billiard, formed by the intersection of two disks of different radii (one with radius 1 and the other with radius r > 1). Although this system does not satisfy the Defocusing Mechanism, it still exhibits hyperbolicity under certain conditions, contradicting traditional geometric intuition and bringing a new perspective on the concept of hyperbolicity and chaos.