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.

[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. doi: 10.1007/s00023-009-0409-9.

[4]

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

[5]

K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley & Sons, (1990).

[6]

D. Genin, "Regular and Chaotic Dynamics of Outer Billiards,", Ph.D. thesis, (2005).

[7]

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

[8]

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

[9]

L. Li, On Moser's boundedness problem of dual billiards,, Ergodic Theorem and Dynamical Systems, 29 (2009), 613. 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, 77 (1973).

[12]

B. H. Neumann, "Sharing Ham and Eggs,", Summary of a Manchester Mathematics Colloquium, (1959).

[13]

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

[14]

R. E. Schwartz, "Outer Billiards on Kites,", Annals of Mathematics Studies, 171 (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 (2005).

[18]

S. Tabachnikov, "Billiards,", Panoramas et Syntheses, 1 (1995).

[19]

F. Vivaldi and A. Shaidenko, Global stability of a class of discontinuous dual billiards,, Comm. Math. Phys., 110 (1987), 625. doi: 10.1007/BF01205552.

show all references

References:
[1]

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

[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. doi: 10.1007/s00023-009-0409-9.

[4]

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

[5]

K. J. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley & Sons, (1990).

[6]

D. Genin, "Regular and Chaotic Dynamics of Outer Billiards,", Ph.D. thesis, (2005).

[7]

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

[8]

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

[9]

L. Li, On Moser's boundedness problem of dual billiards,, Ergodic Theorem and Dynamical Systems, 29 (2009), 613. 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, 77 (1973).

[12]

B. H. Neumann, "Sharing Ham and Eggs,", Summary of a Manchester Mathematics Colloquium, (1959).

[13]

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

[14]

R. E. Schwartz, "Outer Billiards on Kites,", Annals of Mathematics Studies, 171 (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 (2005).

[18]

S. Tabachnikov, "Billiards,", Panoramas et Syntheses, 1 (1995).

[19]

F. Vivaldi and A. Shaidenko, Global stability of a class of discontinuous dual billiards,, Comm. Math. Phys., 110 (1987), 625. doi: 10.1007/BF01205552.

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