Nontrivial paths and periodic orbits of the T-fractal billiard table

Abstract

We introduce and prove numerous new results about the orbits of the T-fractal billiard. Specifically, in section , we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In section , we examine the limiting behavior of particular sequences of compatible periodic orbits. Additionally, sufficient conditions for the existence of particular nontrivial paths are given in section . The proofs of two results of Lapidus and Niemeyer (2013 The current state of fractal billiards Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics (Contemporary Mathematics vol 601) ed D Carfi et al (Providence, RI: American Mathematical Society) pp 251–88 (e-print: arXiv:math.DS.1210.0282v2, 2013) appear here for the first time, as well. In section , an orbit with an irrational initial direction reaches an elusive point in a way that yields a nontrivial path of finite length, yet, by our convention, constitutes a singular orbit of the fractal billiard table. The existence of such an orbit seems to indicate that the classification of orbits may not be so straightforward. A discussion of our results and directions for future research is then given in section .