# American Institute of Mathematical Sciences

August  2013, 33(8): 3719-3740. doi: 10.3934/dcds.2013.33.3719

## Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards

 1 University of California, Riverside, 900 Big Springs Rd., Riverside, CA 92521, United States, United States

Received  April 2012 Revised  October 2012 Published  January 2013

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.
Citation: Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719
##### References:
 [1] J. P. Chen and R. G. Niemeyer, Periodic billiard orbits of self-similar Sierpinski carpets, in Preparation, (2012). [2] E. Durane-Cartagena and J. T. Tyson, Rectifiable curves in Sierpiński carpets, to appear in Indiana Univ. Math. J., (2011). [3] K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Chichester, 1990. (2nd edition, 2003.) doi: 10.1002/0470013850. [4] G. Galperin, Ya. B. Vorobets and A. M. Stepin, Periodic billiard trajectories in polygons, Russian Math. Surveys, 47 (1992), 5-80. doi: 10.1070/RM1992v047n03ABEH000893. [5] E. Gutkin, Billiards in polygons: Survey of recent results, J. Stat. Phys., 83 (1996), 7-26. doi: 10.1007/BF02183637. [6] E. Gutkin, Billiards on almost integrable polyhedral surfaces, Erg. Th. and Dyn. Syst., 4 (1984), 569-584. doi: 10.1017/S0143385700002650. [7] E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., 3 (1996), 391-403. [8] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3. [9] P. Hubert and T. Schmidt, An introduction to Veech surfaces, in "Handbook of Dynamical Systems" 1B (eds. A. Katok and B. Hasselblatt), Elsevier, Amsterdam, (2006), 501-526. doi: 10.1016/S1874-575X(06)80031-7. [10] A. Katok and B. Hasselblatt, "A First Course in Dynamics: With a Panorama of Recent Developments," Cambridge Univ. Press, Cambridge, 2003. [11] A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 760-764. [12] M. L. Lapidus and R. G. Niemeyer, Towards the Koch snowflake fractal billiard-Computer experiments and mathematical conjectures, in "Gems in Experimental Mathematics" (eds. T. Amdeberhan, L. A. Medina and V. H. Moll), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 517 (2010), 231-263. [E-print: arXiv:math.DS.0912.3948v1, 2009.] doi: 10.1090/conm/517/10144. [13] M. L. Lapidus and R. G. Niemeyer, Families of periodic orbits of the Koch snowflake fractal billiard, 63 pages, e-print, arXiv:1105.0737v1, (2011). [14] M. L. Lapidus and R. G. Niemeyer, Veech groups $\Gamma_n$ of the Koch snowflake prefractal flat surfaces $\mathcalS(\ks_n)$, in Progress, 2012. [15] M. L. Lapidus and R. G. Niemeyer, Experimental evidence in support of a fractal law of reflection, in progress, (2012). [16] W. S. Massey, "Algebraic Topology: An Introduction," Springer-Verlag, New York, 1977. [17] H. Masur, Closed trajectories for quadratic differentials with an applications to billiards, Duke Math. J., 53 (1986), 307-314. doi: 10.1215/S0012-7094-86-05319-6. [18] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems" 1A (eds. A. Katok and B. Hasselblatt), Elsevier, Amsterdam, (2002), 1015-1090. doi: 10.1016/S1874-575X(02)80015-7. [19] J. Smillie, Dynamics of billiard flow in rational polygons, in "Dynamical Systems" Encyclopedia of Math. Sciences, 100, Math. Physics 1 ( ed. Ya. G. Sinai), Springer-Verlag, New York, (2000), 360-382. [20] S. Tabachnikov, "Billiards," Panoramas et Synthèses, Soc. Math. France, Paris, 1995. [21] S. Tabachnikov, "Geometry and Billiards," Amer. Math. Soc., Providence, RI, 2005. [22] W. A. Veech, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876. [23] W. A. Veech, Flat surfaces, Amer. J. Math., 115 (1993), 589-689. doi: 10.2307/2375075. [24] Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys, 51 (1996), 779-817. doi: 10.1070/RM1996v051n05ABEH002993. [25] G. Weitze-Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472. [26] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics and Geometry" I (eds. P. Cartier, et al.,), Springer-Verlag, Berlin, (2002), 439-585. doi: 10.1007/978-3-540-30308-4.

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##### References:
 [1] J. P. Chen and R. G. Niemeyer, Periodic billiard orbits of self-similar Sierpinski carpets, in Preparation, (2012). [2] E. Durane-Cartagena and J. T. Tyson, Rectifiable curves in Sierpiński carpets, to appear in Indiana Univ. Math. J., (2011). [3] K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Chichester, 1990. (2nd edition, 2003.) doi: 10.1002/0470013850. [4] G. Galperin, Ya. B. Vorobets and A. M. Stepin, Periodic billiard trajectories in polygons, Russian Math. Surveys, 47 (1992), 5-80. doi: 10.1070/RM1992v047n03ABEH000893. [5] E. Gutkin, Billiards in polygons: Survey of recent results, J. Stat. Phys., 83 (1996), 7-26. doi: 10.1007/BF02183637. [6] E. Gutkin, Billiards on almost integrable polyhedral surfaces, Erg. Th. and Dyn. Syst., 4 (1984), 569-584. doi: 10.1017/S0143385700002650. [7] E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., 3 (1996), 391-403. [8] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3. [9] P. Hubert and T. Schmidt, An introduction to Veech surfaces, in "Handbook of Dynamical Systems" 1B (eds. A. Katok and B. Hasselblatt), Elsevier, Amsterdam, (2006), 501-526. doi: 10.1016/S1874-575X(06)80031-7. [10] A. Katok and B. Hasselblatt, "A First Course in Dynamics: With a Panorama of Recent Developments," Cambridge Univ. Press, Cambridge, 2003. [11] A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 760-764. [12] M. L. Lapidus and R. G. Niemeyer, Towards the Koch snowflake fractal billiard-Computer experiments and mathematical conjectures, in "Gems in Experimental Mathematics" (eds. T. Amdeberhan, L. A. Medina and V. H. Moll), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 517 (2010), 231-263. [E-print: arXiv:math.DS.0912.3948v1, 2009.] doi: 10.1090/conm/517/10144. [13] M. L. Lapidus and R. G. Niemeyer, Families of periodic orbits of the Koch snowflake fractal billiard, 63 pages, e-print, arXiv:1105.0737v1, (2011). [14] M. L. Lapidus and R. G. Niemeyer, Veech groups $\Gamma_n$ of the Koch snowflake prefractal flat surfaces $\mathcalS(\ks_n)$, in Progress, 2012. [15] M. L. Lapidus and R. G. Niemeyer, Experimental evidence in support of a fractal law of reflection, in progress, (2012). [16] W. S. Massey, "Algebraic Topology: An Introduction," Springer-Verlag, New York, 1977. [17] H. Masur, Closed trajectories for quadratic differentials with an applications to billiards, Duke Math. J., 53 (1986), 307-314. doi: 10.1215/S0012-7094-86-05319-6. [18] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems" 1A (eds. A. Katok and B. Hasselblatt), Elsevier, Amsterdam, (2002), 1015-1090. doi: 10.1016/S1874-575X(02)80015-7. [19] J. Smillie, Dynamics of billiard flow in rational polygons, in "Dynamical Systems" Encyclopedia of Math. Sciences, 100, Math. Physics 1 ( ed. Ya. G. Sinai), Springer-Verlag, New York, (2000), 360-382. [20] S. Tabachnikov, "Billiards," Panoramas et Synthèses, Soc. Math. France, Paris, 1995. [21] S. Tabachnikov, "Geometry and Billiards," Amer. Math. Soc., Providence, RI, 2005. [22] W. A. Veech, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876. [23] W. A. Veech, Flat surfaces, Amer. J. Math., 115 (1993), 589-689. doi: 10.2307/2375075. [24] Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Russian Math. Surveys, 51 (1996), 779-817. doi: 10.1070/RM1996v051n05ABEH002993. [25] G. Weitze-Schmithüsen, An algorithm for finding the Veech group of an origami, Experimental Mathematics, 13 (2004), 459-472. [26] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics and Geometry" I (eds. P. Cartier, et al.,), Springer-Verlag, Berlin, (2002), 439-585. doi: 10.1007/978-3-540-30308-4.
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