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

We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we examine the limiting behavior of particular sequences of compatible periodic orbits and, more interesting, in Section 5, the limiting behavior of a particular sequence of compatible singular orbits. The latter seems to indicate that the classification of orbits may not be so straightforward. Additionally, sufficient conditions for the existence of particular nontrivial paths is given in Section 4. The proofs of two results stated in [LapNie4] appear here for the first time, as well. A discussion of our results and directions for future research is then given in Section 6.