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. Google Scholar

[2]

R. Douady, "These de 3-Eme Cycle," Université de Paris 7, 1982. Google Scholar

[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.  Google Scholar

[4]

F. Dogru and S. Tabachnikov, Dual billiards, Math. Intelligencer, 27 (2005), 18-25.  Google Scholar

[5]

K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[6]

D. Genin, "Regular and Chaotic Dynamics of Outer Billiards," Ph.D. thesis, The Pennsylvania State University, 2005.  Google Scholar

[7]

E. Gutkin and N. Simányi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1992), 431-449. doi: 10.1007/BF02099259.  Google Scholar

[8]

R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1989), 163-168.  Google Scholar

[9]

L. Li, On Moser's boundedness problem of dual billiards, Ergodic Theorem and Dynamical Systems, 29 (2009), 613-635. doi: 10.1017/S0143385708000515.  Google Scholar

[10]

J. Moser, Is the solar system stable?,, Math. Intelligencer, 1 (): 65.  doi: 10.1007/BF03023062.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

R. E. Schwartz, Unbounded orbits for outer billiards, J. Mod. Dyn., 1 (2007), 371-424. doi: 10.3934/jmd.2007.1.371.  Google Scholar

[14]

R. E. Schwartz, "Outer Billiards on Kites," Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009.  Google Scholar

[15]

R. E. Schwartz, Outer billiards and the pinwheel map, Journal of Modern Dynamics, 2011. Google Scholar

[16]

R. E. Schwartz, Outer Billiards, Quarter Turn Compositions, and Polytope Exchange Transformations, preprint, 2011. Google Scholar

[17]

S. Tabachnikov, "Geometry and Billiards," Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI, Mathematics Advanced Study Semesters, University Park, PA, 2005.  Google Scholar

[18]

S. Tabachnikov, "Billiards," Panoramas et Syntheses, 1, Société Mathématique de France, 1995. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

N. E. J. De Bruijn, Algebraic theory of Penrose's nonperiodic tilings, Nederl. Akad. Wentensch. Proc., 84 (1981), 39-66. Google Scholar

[2]

R. Douady, "These de 3-Eme Cycle," Université de Paris 7, 1982. Google Scholar

[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.  Google Scholar

[4]

F. Dogru and S. Tabachnikov, Dual billiards, Math. Intelligencer, 27 (2005), 18-25.  Google Scholar

[5]

K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[6]

D. Genin, "Regular and Chaotic Dynamics of Outer Billiards," Ph.D. thesis, The Pennsylvania State University, 2005.  Google Scholar

[7]

E. Gutkin and N. Simányi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1992), 431-449. doi: 10.1007/BF02099259.  Google Scholar

[8]

R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1989), 163-168.  Google Scholar

[9]

L. Li, On Moser's boundedness problem of dual billiards, Ergodic Theorem and Dynamical Systems, 29 (2009), 613-635. doi: 10.1017/S0143385708000515.  Google Scholar

[10]

J. Moser, Is the solar system stable?,, Math. Intelligencer, 1 (): 65.  doi: 10.1007/BF03023062.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

R. E. Schwartz, Unbounded orbits for outer billiards, J. Mod. Dyn., 1 (2007), 371-424. doi: 10.3934/jmd.2007.1.371.  Google Scholar

[14]

R. E. Schwartz, "Outer Billiards on Kites," Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009.  Google Scholar

[15]

R. E. Schwartz, Outer billiards and the pinwheel map, Journal of Modern Dynamics, 2011. Google Scholar

[16]

R. E. Schwartz, Outer Billiards, Quarter Turn Compositions, and Polytope Exchange Transformations, preprint, 2011. Google Scholar

[17]

S. Tabachnikov, "Geometry and Billiards," Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI, Mathematics Advanced Study Semesters, University Park, PA, 2005.  Google Scholar

[18]

S. Tabachnikov, "Billiards," Panoramas et Syntheses, 1, Société Mathématique de France, 1995. Google Scholar

[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.  Google Scholar

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