# American Institute of Mathematical Sciences

July  2011, 5(3): 473-581. doi: 10.3934/jmd.2011.5.473

## Outer billiards on the Penrose kite: Compactification and renormalization

 1 Department of Mathematics, Brown University, Providence, RI 02912, United States

Received  February 2011 Revised  August 2011 Published  November 2011

We give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this $2$-dimensional dynamical system has a $3$-dimensional compactification, a certain polyhedron exchange map defined on the $3$-torus, and that this $3$-dimensional system admits a renormalization scheme. The two features allow us to make sharp statements concerning the distribution, large- and fine-scale geometry, and hidden algebraic symmetry, of the orbits. One concrete result is that the union of the unbounded orbits has Hausdorff dimension $1$. We establish many of the results with computer-aided proofs that involve only integer arithmetic.
Citation: Richard Evan Schwartz. Outer billiards on the Penrose kite: Compactification and renormalization. Journal of Modern Dynamics, 2011, 5 (3) : 473-581. doi: 10.3934/jmd.2011.5.473
##### References:
 [1] N. E. J. De Bruijn, Algebraic theory of Penrose's nonperiodic tilings, Nederl. Akad. Wentensch. Proc., 84 (1981), 39-66. [2] R. Douady, "These de 3-Eme Cycle," Université de Paris 7, 1982. [3] D. Dolyopyat and B. Fayad, Unbounded orbits for semicircular outer billiards, Annales Henri Poincaré, 10 (2009), 357-375. doi: 10.1007/s00023-009-0409-9. [4] F. Dogru and S. Tabachnikov, Dual billiards, Math. Intelligencer, 27 (2005), 18-25. [5] K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Ltd., Chichester, 1990. [6] D. Genin, "Regular and Chaotic Dynamics of Outer Billiards," Ph.D. thesis, The Pennsylvania State University, 2005. [7] E. Gutkin and N. Simányi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1992), 431-449. doi: 10.1007/BF02099259. [8] R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1989), 163-168. [9] L. Li, On Moser's boundedness problem of dual billiards, Ergodic Theorem and Dynamical Systems, 29 (2009), 613-635. doi: 10.1017/S0143385708000515. [10] J. Moser, Is the solar system stable?,, Math. Intelligencer, 1 (): 65.  doi: 10.1007/BF03023062. [11] J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics," Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N.J., Ann. of Math. Stud., 77, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1973. [12] B. H. Neumann, "Sharing Ham and Eggs," Summary of a Manchester Mathematics Colloquium, 25 Jan 1959, published in Iota, the Manchester University Mathematics Students' Journal. [13] R. E. Schwartz, Unbounded orbits for outer billiards, J. Mod. Dyn., 1 (2007), 371-424. doi: 10.3934/jmd.2007.1.371. [14] R. E. Schwartz, "Outer Billiards on Kites," Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009. [15] R. E. Schwartz, Outer billiards and the pinwheel map, Journal of Modern Dynamics, 2011. [16] R. E. Schwartz, Outer Billiards, Quarter Turn Compositions, and Polytope Exchange Transformations, preprint, 2011. [17] S. Tabachnikov, "Geometry and Billiards," Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI, Mathematics Advanced Study Semesters, University Park, PA, 2005. [18] S. Tabachnikov, "Billiards," Panoramas et Syntheses, 1, Société Mathématique de France, 1995. [19] F. Vivaldi and A. Shaidenko, Global stability of a class of discontinuous dual billiards, Comm. Math. Phys., 110 (1987), 625-640. doi: 10.1007/BF01205552.

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##### References:
 [1] N. E. J. De Bruijn, Algebraic theory of Penrose's nonperiodic tilings, Nederl. Akad. Wentensch. Proc., 84 (1981), 39-66. [2] R. Douady, "These de 3-Eme Cycle," Université de Paris 7, 1982. [3] D. Dolyopyat and B. Fayad, Unbounded orbits for semicircular outer billiards, Annales Henri Poincaré, 10 (2009), 357-375. doi: 10.1007/s00023-009-0409-9. [4] F. Dogru and S. Tabachnikov, Dual billiards, Math. Intelligencer, 27 (2005), 18-25. [5] K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Ltd., Chichester, 1990. [6] D. Genin, "Regular and Chaotic Dynamics of Outer Billiards," Ph.D. thesis, The Pennsylvania State University, 2005. [7] E. Gutkin and N. Simányi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1992), 431-449. doi: 10.1007/BF02099259. [8] R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1989), 163-168. [9] L. Li, On Moser's boundedness problem of dual billiards, Ergodic Theorem and Dynamical Systems, 29 (2009), 613-635. doi: 10.1017/S0143385708000515. [10] J. Moser, Is the solar system stable?,, Math. Intelligencer, 1 (): 65.  doi: 10.1007/BF03023062. [11] J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics," Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N.J., Ann. of Math. Stud., 77, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1973. [12] B. H. Neumann, "Sharing Ham and Eggs," Summary of a Manchester Mathematics Colloquium, 25 Jan 1959, published in Iota, the Manchester University Mathematics Students' Journal. [13] R. E. Schwartz, Unbounded orbits for outer billiards, J. Mod. Dyn., 1 (2007), 371-424. doi: 10.3934/jmd.2007.1.371. [14] R. E. Schwartz, "Outer Billiards on Kites," Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009. [15] R. E. Schwartz, Outer billiards and the pinwheel map, Journal of Modern Dynamics, 2011. [16] R. E. Schwartz, Outer Billiards, Quarter Turn Compositions, and Polytope Exchange Transformations, preprint, 2011. [17] S. Tabachnikov, "Geometry and Billiards," Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI, Mathematics Advanced Study Semesters, University Park, PA, 2005. [18] S. Tabachnikov, "Billiards," Panoramas et Syntheses, 1, Société Mathématique de France, 1995. [19] F. Vivaldi and A. Shaidenko, Global stability of a class of discontinuous dual billiards, Comm. Math. Phys., 110 (1987), 625-640. doi: 10.1007/BF01205552.
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