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- W1433446725 abstract "If D is a rational polygon, then the associated rational billiard table is given by (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 (F). In this paper, we survey many of the results from (LapNie1-3) for the Koch snowflake fractal billiard (KS) and announce new results on two other fractal billiard tables, namely, the T -fractal billiard table (T ) (see (LapNie6)) and a self-similar Sierpinski carpet billiard table (Sa) (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 (KSn), (Tn) and (Sa,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 (F) or else a periodic orbit of (F) with finite period. In our examples, F will be either KS, T or Sa. 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." @default.
- W1433446725 created "2016-06-24" @default.
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- W1433446725 date "2013-01-01" @default.
- W1433446725 modified "2023-10-18" @default.
- W1433446725 title "The Current State of Fractal Billiards" @default.
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- W1433446725 doi "https://doi.org/10.1090/conm/601/11956" @default.
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