Families of Periodic Orbits of the Koch Snowflake Fractal Billiard

Details Families of Periodic Orbits of the Koch Snowflake Fractal Billiard Families of Periodic Orbits of the Koch Snowflake Fractal Billiard

2011-05-04

arxiv.org/abs/1105.0737v1

Mathematics

Dynamical Systems

This paper contains an index of notation, a table of contents and 31 figures; it is 63 pages in length

Scientific paper

We describe the periodic orbits of the prefractal Koch snowflake billiard (the nth inner rational polygonal approximation of the Koch snowflake billiard). In the case of the finite (prefractal) billiard table, we focus on the direction given by an initial angle of pi/3, and define 1) a compatible sequence of piecewise Fagnano orbits, 2) an eventually constant compatible sequence of orbits and 3) a compatible sequence of generalized piecewise Fagnano orbits. In the case of the infinite (fractal) billiard table, we will describe what we call stabilizing periodic orbits of the Koch snowflake fractal billiard. In a sense, we show that it is possible to define billiard dynamics on a Cantor set. In addition, we will show that the inverse limit of the footprints of orbits of the prefractal approximations exists in a specific situation and provide a plausibility argument as to why such an inverse limit of footprints should constitute the footprint of a well-defined periodic orbit of the fractal billiard. Using known results for the inverse limit of a sequence of finite spaces, we deduce that the footprint (i.e., the intersection of the orbit with the boundary) of a piecewise Fagnano orbit is a topological Cantor set and a self-similar Cantor set. We allude to a possible characterization of orbits with an initial direction of pi/3. Such a characterization would allow one to describe an orbit with an initial direction of pi/3 of the Koch snowflake billiard as either a piecewise Fagnano orbit, a stabilizing orbit or a generalized piecewise Fagnano orbit. We then close the paper by discussing several outstanding open problems and conjectures about the Koch snowflake fractal billiard, the associated 'fractal flat surface', and possible connections with the associated fractal drum. In the long-term, the present work may help lay the foundations for a general theory of fractal billiards.

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