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April  2011, 5(2): 255-283. doi: 10.3934/jmd.2011.5.255

Outer billiards and the pinwheel map

Richard Evan Schwartz 1,

1. 

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

Received  July 2010 Revised  March 2011 Published  July 2011

In this paper we establish an equivalence between an outer billiards system based on a convex polygon $P$ and an auxiliary system, which we call the pinwheel map, that is based on $P$ in a different way. The pinwheel map is akin to a first-return map of the outer billiards map. The virtue of our result is that most of the main questions about outer billiards can be formulated in terms of the pinwheel map, and the pinwheel map is simpler and seems more amenable to fruitful analysis.
Keywords: piecewise translations, pinwheel map., Outer billiards.
Mathematics Subject Classification: Primary: 37E15; Secondary: 37E9.
Citation: Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
References:
[1]

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

[2]

D. Dolyopyat and B. Fayad, Unbounded orbits for semicircular outer billiards,, Annales Henri Poincaré, ().   Google Scholar

[3]

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

[4]

F. Dogru and S. Tabachnikov, Dual billiards in the hyperbolic plane, Nonlinearity, 15 (2002), 1051-1072. doi: 10.1088/0951-7715/15/4/305.  Google Scholar

[5]

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

[6]

E. Gutkin and N. Simanyi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1991), 431-450. doi: 10.1007/BF02099259.  Google Scholar

[7]

R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1994), 163-168.  Google Scholar

[8]

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

[9]

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. Annals of Mathematics Studies, No. 77. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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]

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

[2]

D. Dolyopyat and B. Fayad, Unbounded orbits for semicircular outer billiards,, Annales Henri Poincaré, ().   Google Scholar

[3]

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

[4]

F. Dogru and S. Tabachnikov, Dual billiards in the hyperbolic plane, Nonlinearity, 15 (2002), 1051-1072. doi: 10.1088/0951-7715/15/4/305.  Google Scholar

[5]

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

[6]

E. Gutkin and N. Simanyi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1991), 431-450. doi: 10.1007/BF02099259.  Google Scholar

[7]

R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1994), 163-168.  Google Scholar

[8]

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

[9]

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. Annals of Mathematics Studies, No. 77. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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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