Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards

Pages: 3719 - 3740, Volume 33, Issue 8, August 2013      doi:10.3934/dcds.2013.33.3719

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Michel L. Lapidus - University of California, Riverside, 900 Big Springs Rd., Riverside, CA 92521, United States (email)
Robert G. Niemeyer - University of California, Riverside, 900 Big Springs Rd., Riverside, CA 92521, United States (email)

Abstract: The Koch snowflake $KS$ is a nowhere differentiable curve. The billiard table $Ω (KS)$ with boundary $KS$ is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes $Ω (KS)$ such an interesting, yet difficult, table to analyze.
    In this paper, we approach this problem by approximating (from the inside) $Ω (KS)$ by well-defined (prefractal) rational polygonal billiard tables $Ω (KS_{n})$. We first show that the flat surface $S(KS_{n})$ determined from the rational billiard $Ω (KS_{n})$ is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [6], allows us to define a sequence of compatible orbits of prefractal billiards $Ω (KS_{n})$. Using a particular addressing system, we define a hybrid orbit of a prefractal billiard $Ω (KS_{n})$ and show that every dense orbit of a prefractal billiard $Ω (KS_{n})$ is a dense hybrid orbit of $Ω (KS_{n})$. This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits.
    We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of $Ω(KS)$. In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. We conjecture that such a path is indeed the subset of what will eventually be an orbit of the Koch snowflake fractal billiard, once an appropriate `fractal law of reflection' is determined.
    Finally, we close with a discussion of several open problems and potential directions for further research. We discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining 'fractal flat surfaces' naturally associated with the billiard flows.

Keywords:  Fractal billiard, polygonal billiard, rational (polygonal) billiard, law of reflection, unfolding process, flat surface, translation surface, geodesic flow, billiard flow, iterated function system and attractor, self-similar set, fractal, prefractal approximations, Koch snowflake billiard, prefractal rational billiard approximations, sequence of compatible orbits, hook orbits, (eventually) constant sequences of compatible orbits, footprints, Cantor points, smooth points, elusive points, periodic orbits, periodic vs. dense orbits.
Mathematics Subject Classification:  Primary: 28A80, 37D40, 37D50, Secondary: 28A75, 37C27, 37E35, 37F40, 58J99.

Received: April 2012;      Revised: October 2012;      Available Online: January 2013.