The current state of fractal billiards

TL;DR

This paper surveys recent results and introduces new findings on the dynamics of billiard tables with fractal boundaries, focusing on the Koch snowflake, T-fractal, and Sierpinski carpet, and analyzes the behavior of compatible orbits and their limits.

Contribution

It provides a unified framework for analyzing sequences of compatible orbits in fractal billiards and presents new results on T-fractal and Sierpinski carpet billiards.

Findings

Identified limiting behaviors of compatible orbits in fractal billiards.

Established existence of nontrivial paths and periodic orbits in fractal tables.

Presented first-time examples and results for T-fractal and Sierpinski carpet billiards.

Abstract

If D is a rational polygon, then the associated rational billiard table is given by \Omega(D). Such a billiard table is well understood. If F is a closed fractal curve approximated by a sequence of rational polygons, then the corresponding fractal billiard table is denoted by \Omega(F). In this paper, we survey many of the results from [LapNie1-3] for the Koch snowflake fractal billiard \Omega(KS) and announce new results on two other fractal billiard tables, namely, the T-fractal billiard table \Omega(T) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table \Omega(S_a) (see [CheNie]). We build a general framework within which to analyze what we call a sequence of compatible orbits. Properties of particular sequences of compatible orbits are discussed for each prefractal billiard \Omega(KS_n), \Omega(T_n) and \Omega(S_a,n), for n = 0, 1, 2... . In each case, we are able to determine a particular limiting behavior for an appropriately formulated sequence of compatible orbits. Such a limit either constitutes what we call a nontrivial path of a fractal billiard table \Omega(F) or else a periodic orbit of \Omega(F) with finite period. In our examples, F will be either KS, T or S_a. Several of the results and examples discussed in this paper are presented for the first time. We then close with a brief discussion of open problems and directions for further research in the emerging field of fractal billiards.