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Towards the Koch snowflake fractal billiard: computer experiments and mathematical conjectures. (English) Zbl 1222.37028

Amdeberhan, Tewodros (ed.) et al., Gems in experimental mathematics. AMS special session on experimental mathematics, Washington, DC, January 5, 2009. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4869-2/pbk). Contemporary Mathematics 517, 231-263 (2010).
This article is part of a program to understand the Koch snowflake. (For more details on this topic see, e.g., [M. L. Lapidus and M. M. H. Pang, Commun. Math. Phys. 172, No. 2, 359–376 (1995; Zbl 0857.35093); M. L. Lapidus et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, No. 7, 1185–1210 (1996; Zbl 0920.73165); C. A. Griffith and M. L. Lapidus, Computer graphics and the eigenfunctions for the Koch snowflake drum. Basel: Birkhäuser. Trends in Mathematics, 95–113 (1997; Zbl 1044.58504); M. L. Lapidus and E. P. J. Pearse, J. Lond. Math. Soc., II. Ser. 74, No. 2, 397–414 (2006; Zbl 1110.26006); B. Daudert and M. L. Lapidus, Fractals 15, No. 3, 255–272 (2007; Zbl 1137.28301)]). Here the authors attempt to define some orbits of the (still to be defined) Koch snowflake billiard flow.
Since the boundary of the Koch snowflake is nowhere differentiable, it is a priori not at all clear what it should mean to reflect a billiard ball at this boundary. The authors propose to take advantage of the construction of the Koch snowflake as the limit of the well-known sequence of rational polygons, starting at the equilateral triangle. The billiard flow on the equilateral triangle is perfectly understood. They attempt to define orbits and the (fractal) billiard flow of the Koch snowflake as a limit (in a still-to-be-defined sense) of the orbits and the billiard flows on this sequence.
To that end they investigate orbits of these rational billiard flows and define the notion of induced orbits. Mainly they study Fagnano orbits (shortest length) and variants, as well as quasiperiodic orbits. They close with a list of well-motivated conjectures and open problems. This paper is well-written and contains many illustrating figures.
For the entire collection see [Zbl 1193.00060].

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37C27 Periodic orbits of vector fields and flows
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65P99 Numerical problems in dynamical systems
37A99 Ergodic theory
37C55 Periodic and quasi-periodic flows and diffeomorphisms
58A99 General theory of differentiable manifolds
74H99 Dynamical problems in solid mechanics
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