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January  2014, 8(1): 109-132. doi: 10.3934/jmd.2014.8.109

Pseudo-integrable billiards and arithmetic dynamics

Vladimir Dragović 1, and  Milena Radnović 2,

1. 

Department of Mathematical Sciences, University of Texas at Dallas, FO 35, 800West Campbell Road, TX 75080 USA, Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia
2. 

Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia

Received  September 2013 Published  July 2014

We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behavior different from Liouville--Arnold's Theorem. Its analog is derived from the Maier Theorem on measured foliations. The billiard flow generates a measurable foliation defined by a closed 1-form $w$. Using the closed form, a transformation of the given billiard table to a rectangular cylinder is constructed and a trajectory equivalence between corresponding billiards has been established. A local version of Poncelet Theorem is formulated and necessary algebro-geometric conditions for periodicity are presented. It is proved that the dynamics depends on arithmetic of rotation numbers, but not on geometry of a given confocal pencil of conics.
Keywords: polygonal billiards, Poncelet theorem, measured foliations., periodic billiard trajectories, interval exchange, Confocal conics.
Mathematics Subject Classification: Primary: 37D50, 37J35, 37A05; Secondary: 28D0.
Citation: Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109
References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer Verlag, New York-Heidelberg, 1978.  Google Scholar

[2]

V. I. Arnold, Poly-integrable flows, (Russian) Algebra i Analiz, 4 (1992), 54-62; translation in St. Petersburg Math. J., 4 (1993), 1103-1110.  Google Scholar

[3]

J. S. Athreya, A. Eskin and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbb CP^1$, arXiv:1212.1660, 2013. Google Scholar

[4]

H. J. M. Bos, C. Kers, F. Oort and D. W. Raven, Poncelet's closure theorem, Expo. Math., 5 (1987), 289-364.  Google Scholar

[5]

A. Cayley, Note on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 6 (1853), 99-102. Google Scholar

[6]

A. Cayley, Developments on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 7 (1854), 339-345. Google Scholar

[7]

M. Chasles, Géométrie pure. Théorèmes sur les sections coniques confocales,, Ann. Math. Pures Appl. [Ann. Gergonne], 18 (): 269.   Google Scholar

[8]

G. Darboux, Sur les polygones inscrits et circonscrits à l'ellipsoĩde, Bulletin de la Société Philomathique, 7 (1870), 92-94. Google Scholar

[9]

V. Dragović and M. Radnović, Cayley-type conditions for billiards within $k$ quadrics in $\mathbf R^d$, J. Phys. A, 37 (2004), 1269-1276. doi: 10.1088/0305-4470/37/4/014.  Google Scholar

[10]

V. Dragović and M. Radnović, A survey of the analytical description of periodic elliptical billiard trajectories, J. Math. Sci. (N. Y.), 135 (2006), 3244-3255. doi: 10.1007/s10958-006-0154-2.  Google Scholar

[11]

V. Dragović and M. Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl. (9), 85 (2006), 758-790. doi: 10.1016/j.matpur.2005.12.002.  Google Scholar

[12]

V. Dragović and M. Radnović, Bifurcations of Liouville tori in elliptical billiards, Regul. Chaotic Dyn., 14 (2009), 479-494. doi: 10.1134/S1560354709040054.  Google Scholar

[13]

V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0015-0.  Google Scholar

[14]

J. Fay, Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc., 96 (1992), vi+123 pp. doi: 10.1090/memo/0464.  Google Scholar

[15]

L. Flatto, Poncelet's Theorem, Chapter 15 by S. Tabachnikov, American Mathematical Society, Providence, RI, 2009.  Google Scholar

[16]

C. Jacobi, Fundamenta Nova Theoriae Functiorum Ellipticarum,, 1829., ().   Google Scholar

[17]

C. Jacobi, Gesammelte Werke: Vorlesungen über Dynamic. Supplementband, Berlin, 1884. Google Scholar

[18]

J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609-628. doi: 10.2307/2974690.  Google Scholar

[19]

V. Kozlov and D. Treshchëv, Billiards, Amer. Math. Soc., Providence, RI, 1991.  Google Scholar

[20]

V. V. Kozlov, Dynamical systems on a torus with multivalued integrals, (Russian) Tr. Mat. Inst. Steklova, 256 (2007), Din. Sist. i Optim., 201-218; translation in Proc. Steklov Inst. Math., 256 (2007), 188-205. doi: 10.1134/S0081543807010105.  Google Scholar

[21]

M. Levi and S. Tabachnikov, The Poncelet grid and billiards in ellipses, Amer. Math. Monthly, 114 (2007), 895-908.  Google Scholar

[22]

A. G. Maier, Trajectories on closable orientable surfaces, (Russian) Rec. Math. [Mat. Sbornik] N.S., 12(54) (1943), 71-84.  Google Scholar

[23]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[24]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.  Google Scholar

[25]

J. V. Poncelet, Traité des Propriétés Projectives des Figures, Mett, Paris, 1822. Google Scholar

[26]

P. J. Richens and M. V. Berry, Pseudointegrable systems in classical and quantum mechanics, Physica D, 2 (1981), 495-512. doi: 10.1016/0167-2789(81)90024-5.  Google Scholar

[27]

R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175. doi: 10.1515/ADVGEOM.2007.010.  Google Scholar

[28]

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

[29]

W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl Theorem mod 2, Trans. Amer. Math. Soc., 140 (1969), 1-33.  Google Scholar

[30]

M. Viana, Dynamics of Interval Exchange Maps and Teichmüller flows, Lecture Notes, 2008. Available from: http://w3.impa.br/~viana/out/ietf.pdf. Google Scholar

[31]

A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300.  Google Scholar

[32]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics and Geometry. 1 (eds. P. Cartier, B. Julia, P. Moussa and P. Vanhove), Springer, Berlin, 2006, 437-583. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer Verlag, New York-Heidelberg, 1978.  Google Scholar

[2]

V. I. Arnold, Poly-integrable flows, (Russian) Algebra i Analiz, 4 (1992), 54-62; translation in St. Petersburg Math. J., 4 (1993), 1103-1110.  Google Scholar

[3]

J. S. Athreya, A. Eskin and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbb CP^1$, arXiv:1212.1660, 2013. Google Scholar

[4]

H. J. M. Bos, C. Kers, F. Oort and D. W. Raven, Poncelet's closure theorem, Expo. Math., 5 (1987), 289-364.  Google Scholar

[5]

A. Cayley, Note on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 6 (1853), 99-102. Google Scholar

[6]

A. Cayley, Developments on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 7 (1854), 339-345. Google Scholar

[7]

M. Chasles, Géométrie pure. Théorèmes sur les sections coniques confocales,, Ann. Math. Pures Appl. [Ann. Gergonne], 18 (): 269.   Google Scholar

[8]

G. Darboux, Sur les polygones inscrits et circonscrits à l'ellipsoĩde, Bulletin de la Société Philomathique, 7 (1870), 92-94. Google Scholar

[9]

V. Dragović and M. Radnović, Cayley-type conditions for billiards within $k$ quadrics in $\mathbf R^d$, J. Phys. A, 37 (2004), 1269-1276. doi: 10.1088/0305-4470/37/4/014.  Google Scholar

[10]

V. Dragović and M. Radnović, A survey of the analytical description of periodic elliptical billiard trajectories, J. Math. Sci. (N. Y.), 135 (2006), 3244-3255. doi: 10.1007/s10958-006-0154-2.  Google Scholar

[11]

V. Dragović and M. Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl. (9), 85 (2006), 758-790. doi: 10.1016/j.matpur.2005.12.002.  Google Scholar

[12]

V. Dragović and M. Radnović, Bifurcations of Liouville tori in elliptical billiards, Regul. Chaotic Dyn., 14 (2009), 479-494. doi: 10.1134/S1560354709040054.  Google Scholar

[13]

V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0015-0.  Google Scholar

[14]

J. Fay, Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc., 96 (1992), vi+123 pp. doi: 10.1090/memo/0464.  Google Scholar

[15]

L. Flatto, Poncelet's Theorem, Chapter 15 by S. Tabachnikov, American Mathematical Society, Providence, RI, 2009.  Google Scholar

[16]

C. Jacobi, Fundamenta Nova Theoriae Functiorum Ellipticarum,, 1829., ().   Google Scholar

[17]

C. Jacobi, Gesammelte Werke: Vorlesungen über Dynamic. Supplementband, Berlin, 1884. Google Scholar

[18]

J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609-628. doi: 10.2307/2974690.  Google Scholar

[19]

V. Kozlov and D. Treshchëv, Billiards, Amer. Math. Soc., Providence, RI, 1991.  Google Scholar

[20]

V. V. Kozlov, Dynamical systems on a torus with multivalued integrals, (Russian) Tr. Mat. Inst. Steklova, 256 (2007), Din. Sist. i Optim., 201-218; translation in Proc. Steklov Inst. Math., 256 (2007), 188-205. doi: 10.1134/S0081543807010105.  Google Scholar

[21]

M. Levi and S. Tabachnikov, The Poncelet grid and billiards in ellipses, Amer. Math. Monthly, 114 (2007), 895-908.  Google Scholar

[22]

A. G. Maier, Trajectories on closable orientable surfaces, (Russian) Rec. Math. [Mat. Sbornik] N.S., 12(54) (1943), 71-84.  Google Scholar

[23]

H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[24]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.  Google Scholar

[25]

J. V. Poncelet, Traité des Propriétés Projectives des Figures, Mett, Paris, 1822. Google Scholar

[26]

P. J. Richens and M. V. Berry, Pseudointegrable systems in classical and quantum mechanics, Physica D, 2 (1981), 495-512. doi: 10.1016/0167-2789(81)90024-5.  Google Scholar

[27]

R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175. doi: 10.1515/ADVGEOM.2007.010.  Google Scholar

[28]

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

[29]

W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl Theorem mod 2, Trans. Amer. Math. Soc., 140 (1969), 1-33.  Google Scholar

[30]

M. Viana, Dynamics of Interval Exchange Maps and Teichmüller flows, Lecture Notes, 2008. Available from: http://w3.impa.br/~viana/out/ietf.pdf. Google Scholar

[31]

A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300.  Google Scholar

[32]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics and Geometry. 1 (eds. P. Cartier, B. Julia, P. Moussa and P. Vanhove), Springer, Berlin, 2006, 437-583. doi: 10.1007/978-3-540-31347-2_13.  Google Scholar

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