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

TL;DR

This paper investigates the complex behavior of orbits in the $T$-fractal billiard table, providing new conditions for periodic orbits, analyzing their limits, and exploring nontrivial paths, revealing intricate orbit classifications.

Contribution

It introduces new sufficient conditions for compatible periodic orbits and analyzes their limiting behavior, advancing understanding of orbit dynamics in fractal billiards.

Findings

Sufficient conditions for existence of compatible periodic orbits

Analysis of limiting behavior of sequences of orbits

Identification of complex orbit classifications

Abstract

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.