Undergraduate honors thesis on the Birkhoff conjecture for convex billiards. Contains mathematica notebooks used to visualize the dynamics in near-elliptical billiards and calculate Lyapunov exponents, LateX code for honors thesis presentation, LateX code for the actual thesis. Please refer to the thesis pdf for a concise description of the thesis project and background information.
Abstract: The Birkhoff conjecture claims that if a convex billiard is integrable, then the boundary is necessarily an ellipse. In this paper, we analyze a local version of this conjecture that a small integrable perturbation of an ellipse must be an ellipse as proven by Kaloshin and Sorrentino. As an introduction, we discuss the behavior of billiards in a disc and in an ellipse. We then move onto some of the properties of a near-elliptical billiard. The dynamics of this system are modeled by a two-dimensional nonlinear mapping. The phase space of such billiards are of mixed kind with chaotic and regular behavior. We conclude by analyzing the behavior of the positive Lyapunov exponent and integrability of our system.