# American Institute of Mathematical Sciences

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

## Outer billiards and the pinwheel map

 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.
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. [2] D. Dolyopyat and B. Fayad, Unbounded orbits for semicircular outer billiards,, Annales Henri Poincaré, (). [3] F. Dogru and S. Tabachnikov, Dual billiards, Math. Intelligencer, 27 (2005), 18-25. [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. [5] D. Genin, "Regular and Chaotic Dynamics of Outer Billiards," Ph.D. thesis, The Pennsylvania State University, State College, 2005. [6] E. Gutkin and N. Simanyi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1991), 431-450. doi: 10.1007/BF02099259. [7] R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1994), 163-168. [8] J. Moser, Is the solar system stable?,, Math. Intelligencer, 1 (): 65.  doi: 10.1007/BF03023062. [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. [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. [11] R. E. Schwartz, Unbounded orbits for outer billiards, J. Mod. Dyn., 1 (2007), 371-424. doi: 10.3934/jmd.2007.1.371. [12] R. E. Schwartz, "Outer Billiards on Kites," Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009. [13] S. Tabachnikov, "Geometry and Billiards," Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. [14] S. Tabachnikov, "Billiards," Société Mathématique de France, Panoramas et Syntheses, 1, 1995. [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.

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##### References:
 [1] R. Douady, "These de 3-ème Cycle," Université de Paris 7, 1982. [2] D. Dolyopyat and B. Fayad, Unbounded orbits for semicircular outer billiards,, Annales Henri Poincaré, (). [3] F. Dogru and S. Tabachnikov, Dual billiards, Math. Intelligencer, 27 (2005), 18-25. [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. [5] D. Genin, "Regular and Chaotic Dynamics of Outer Billiards," Ph.D. thesis, The Pennsylvania State University, State College, 2005. [6] E. Gutkin and N. Simanyi, Dual polygonal billiard and necklace dynamics, Comm. Math. Phys., 143 (1991), 431-450. doi: 10.1007/BF02099259. [7] R. Kolodziej, The antibilliard outside a polygon, Bull. Pol. Acad Sci. Math., 37 (1994), 163-168. [8] J. Moser, Is the solar system stable?,, Math. Intelligencer, 1 (): 65.  doi: 10.1007/BF03023062. [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. [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. [11] R. E. Schwartz, Unbounded orbits for outer billiards, J. Mod. Dyn., 1 (2007), 371-424. doi: 10.3934/jmd.2007.1.371. [12] R. E. Schwartz, "Outer Billiards on Kites," Annals of Mathematics Studies, 171, Princeton University Press, Princeton, NJ, 2009. [13] S. Tabachnikov, "Geometry and Billiards," Student Mathematical Library, 30, Amer. Math. Soc., Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. [14] S. Tabachnikov, "Billiards," Société Mathématique de France, Panoramas et Syntheses, 1, 1995. [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.
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