Seminar 11.7.13

In this talk, we will discuss recent results on three different fractal billiard tables: the Koch snowflake, a self-similar Sierpinski carpet and (for lack of a better name and none ever given in the literature) the T-fractal. Initially, an investigation of the flow on a fractal billiard table was made on the Koch snowflake.  Results on the Koch snowflake motivated us to investigate the other two billiard tables. Using a result of J. Tyson and E. Durand-Cartagena, we show that there are periodic orbits of a self-similar Sierpinski carpet billiard table.  J. P. Chen and I have begun investigating whether or not it makes sense to discuss the existence of a dense orbit of a self-similar Sierpinski carpet billiard table. Substantial progress has been made in determining periodic orbits of the T-fractal billiard table.  We detail some of the recent results concerning periodic orbits, determined in collaboration with M. L. Lapidus and R. L. Miller.  Less has been done to determine what may constitute a dense orbit of the T-fractal billiard.  We provide substantial experimental and theoretical evidence in support of the existence of an orbit that is dense in the T-fractal billiard table but is not a space-filling curve. We briefly touch on a long-term goal of determining a topological dichotomy for the flow on a fractal billiard table, namely that, in a fixed direction, the flow is either closed or minimal. Parts of this talk will be suitable for an advanced undergraduate/beginning graduate student audience.