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 & Continuous Dynamical Systems - A, 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). Google Scholar

[2]

E. Durane-Cartagena and J. T. Tyson, Rectifiable curves in Sierpiński carpets,, to appear in Indiana Univ. Math. J., (2011). Google Scholar

[3]

K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley & Sons, (1990). doi: 10.1002/0470013850. Google Scholar

[4]

G. Galperin, Ya. B. Vorobets and A. M. Stepin, Periodic billiard trajectories in polygons,, Russian Math. Surveys, 47 (1992), 5. doi: 10.1070/RM1992v047n03ABEH000893. Google Scholar

[5]

E. Gutkin, Billiards in polygons: Survey of recent results,, J. Stat. Phys., 83 (1996), 7. doi: 10.1007/BF02183637. Google Scholar

[6]

E. Gutkin, Billiards on almost integrable polyhedral surfaces,, Erg. Th. and Dyn. Syst., 4 (1984), 569. doi: 10.1017/S0143385700002650. Google Scholar

[7]

E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards,, Math. Res. Lett., 3 (1996), 391. Google Scholar

[8]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar

[9]

P. Hubert and T. Schmidt, An introduction to Veech surfaces,, in, 1B (2006), 501. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar

[10]

A. Katok and B. Hasselblatt, "A First Course in Dynamics: With a Panorama of Recent Developments,", Cambridge Univ. Press, (2003). Google Scholar

[11]

A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons,, Math. Notes, 18 (1975), 760. Google Scholar

[12]

M. L. Lapidus and R. G. Niemeyer, Towards the Koch snowflake fractal billiard-Computer experiments and mathematical conjectures,, in, 517 (2010), 231. doi: 10.1090/conm/517/10144. Google Scholar

[13]

M. L. Lapidus and R. G. Niemeyer, Families of periodic orbits of the Koch snowflake fractal billiard,, 63 pages, (2011). Google Scholar

[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). Google Scholar

[15]

M. L. Lapidus and R. G. Niemeyer, Experimental evidence in support of a fractal law of reflection,, in progress, (2012). Google Scholar

[16]

W. S. Massey, "Algebraic Topology: An Introduction,", Springer-Verlag, (1977). Google Scholar

[17]

H. Masur, Closed trajectories for quadratic differentials with an applications to billiards,, Duke Math. J., 53 (1986), 307. doi: 10.1215/S0012-7094-86-05319-6. Google Scholar

[18]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, 1A (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar

[19]

J. Smillie, Dynamics of billiard flow in rational polygons,, in, 100 (2000), 360. Google Scholar

[20]

S. Tabachnikov, "Billiards,", Panoramas et Synthèses, (1995). Google Scholar

[21]

S. Tabachnikov, "Geometry and Billiards,", Amer. Math. Soc., (2005). Google Scholar

[22]

W. A. Veech, The billiard in a regular polygon,, Geom. Funct. Anal., 2 (1992), 341. doi: 10.1007/BF01896876. Google Scholar

[23]

W. A. Veech, Flat surfaces,, Amer. J. Math., 115 (1993), 589. doi: 10.2307/2375075. Google Scholar

[24]

Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative,, Russian Math. Surveys, 51 (1996), 779. doi: 10.1070/RM1996v051n05ABEH002993. Google Scholar

[25]

G. Weitze-Schmithüsen, An algorithm for finding the Veech group of an origami,, Experimental Mathematics, 13 (2004), 459. Google Scholar

[26]

A. Zorich, Flat surfaces,, in, I (2002), 439. doi: 10.1007/978-3-540-30308-4. Google Scholar

show all references

References:
[1]

J. P. Chen and R. G. Niemeyer, Periodic billiard orbits of self-similar Sierpinski carpets,, in Preparation, (2012). Google Scholar

[2]

E. Durane-Cartagena and J. T. Tyson, Rectifiable curves in Sierpiński carpets,, to appear in Indiana Univ. Math. J., (2011). Google Scholar

[3]

K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley & Sons, (1990). doi: 10.1002/0470013850. Google Scholar

[4]

G. Galperin, Ya. B. Vorobets and A. M. Stepin, Periodic billiard trajectories in polygons,, Russian Math. Surveys, 47 (1992), 5. doi: 10.1070/RM1992v047n03ABEH000893. Google Scholar

[5]

E. Gutkin, Billiards in polygons: Survey of recent results,, J. Stat. Phys., 83 (1996), 7. doi: 10.1007/BF02183637. Google Scholar

[6]

E. Gutkin, Billiards on almost integrable polyhedral surfaces,, Erg. Th. and Dyn. Syst., 4 (1984), 569. doi: 10.1017/S0143385700002650. Google Scholar

[7]

E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards,, Math. Res. Lett., 3 (1996), 391. Google Scholar

[8]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar

[9]

P. Hubert and T. Schmidt, An introduction to Veech surfaces,, in, 1B (2006), 501. doi: 10.1016/S1874-575X(06)80031-7. Google Scholar

[10]

A. Katok and B. Hasselblatt, "A First Course in Dynamics: With a Panorama of Recent Developments,", Cambridge Univ. Press, (2003). Google Scholar

[11]

A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons,, Math. Notes, 18 (1975), 760. Google Scholar

[12]

M. L. Lapidus and R. G. Niemeyer, Towards the Koch snowflake fractal billiard-Computer experiments and mathematical conjectures,, in, 517 (2010), 231. doi: 10.1090/conm/517/10144. Google Scholar

[13]

M. L. Lapidus and R. G. Niemeyer, Families of periodic orbits of the Koch snowflake fractal billiard,, 63 pages, (2011). Google Scholar

[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). Google Scholar

[15]

M. L. Lapidus and R. G. Niemeyer, Experimental evidence in support of a fractal law of reflection,, in progress, (2012). Google Scholar

[16]

W. S. Massey, "Algebraic Topology: An Introduction,", Springer-Verlag, (1977). Google Scholar

[17]

H. Masur, Closed trajectories for quadratic differentials with an applications to billiards,, Duke Math. J., 53 (1986), 307. doi: 10.1215/S0012-7094-86-05319-6. Google Scholar

[18]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, 1A (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar

[19]

J. Smillie, Dynamics of billiard flow in rational polygons,, in, 100 (2000), 360. Google Scholar

[20]

S. Tabachnikov, "Billiards,", Panoramas et Synthèses, (1995). Google Scholar

[21]

S. Tabachnikov, "Geometry and Billiards,", Amer. Math. Soc., (2005). Google Scholar

[22]

W. A. Veech, The billiard in a regular polygon,, Geom. Funct. Anal., 2 (1992), 341. doi: 10.1007/BF01896876. Google Scholar

[23]

W. A. Veech, Flat surfaces,, Amer. J. Math., 115 (1993), 589. doi: 10.2307/2375075. Google Scholar

[24]

Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative,, Russian Math. Surveys, 51 (1996), 779. doi: 10.1070/RM1996v051n05ABEH002993. Google Scholar

[25]

G. Weitze-Schmithüsen, An algorithm for finding the Veech group of an origami,, Experimental Mathematics, 13 (2004), 459. Google Scholar

[26]

A. Zorich, Flat surfaces,, in, I (2002), 439. doi: 10.1007/978-3-540-30308-4. Google Scholar

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