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

Pseudo-integrable billiards and arithmetic dynamics

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.
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, (1978).

[2]

V. I. Arnold, Poly-integrable flows,, (Russian) Algebra i Analiz, 4 (1992), 54.

[3]

J. S. Athreya, A. Eskin and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbb CP^1$,, , (2013).

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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. doi: 10.1007/s10958-006-0154-2.

[11]

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

[12]

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

[13]

V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics,, Frontiers in Mathematics, (2011). doi: 10.1007/978-3-0348-0015-0.

[14]

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

[15]

L. Flatto, Poncelet's Theorem,, Chapter 15 by S. Tabachnikov, (2009).

[16]

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

[17]

C. Jacobi, Gesammelte Werke: Vorlesungen über Dynamic. Supplementband,, Berlin, (1884).

[18]

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

[19]

V. Kozlov and D. Treshchëv, Billiards,, Amer. Math. Soc., (1991).

[20]

V. V. Kozlov, Dynamical systems on a torus with multivalued integrals,, (Russian) Tr. Mat. Inst. Steklova, 256 (2007), 201. doi: 10.1134/S0081543807010105.

[21]

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

[22]

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

[23]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7.

[24]

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

[25]

J. V. Poncelet, Traité des Propriétés Projectives des Figures,, Mett, (1822).

[26]

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

[27]

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

[28]

S. Tabachnikov, Geometry and Billiards,, Student Mathematical Library, (2005).

[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.

[30]

M. Viana, Dynamics of Interval Exchange Maps and Teichmüller flows,, Lecture Notes, (2008).

[31]

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

[32]

A. Zorich, Flat surfaces,, in Frontiers in Number Theory, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Graduate Texts in Mathematics, (1978).

[2]

V. I. Arnold, Poly-integrable flows,, (Russian) Algebra i Analiz, 4 (1992), 54.

[3]

J. S. Athreya, A. Eskin and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbb CP^1$,, , (2013).

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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. doi: 10.1007/s10958-006-0154-2.

[11]

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

[12]

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

[13]

V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics,, Frontiers in Mathematics, (2011). doi: 10.1007/978-3-0348-0015-0.

[14]

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

[15]

L. Flatto, Poncelet's Theorem,, Chapter 15 by S. Tabachnikov, (2009).

[16]

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

[17]

C. Jacobi, Gesammelte Werke: Vorlesungen über Dynamic. Supplementband,, Berlin, (1884).

[18]

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

[19]

V. Kozlov and D. Treshchëv, Billiards,, Amer. Math. Soc., (1991).

[20]

V. V. Kozlov, Dynamical systems on a torus with multivalued integrals,, (Russian) Tr. Mat. Inst. Steklova, 256 (2007), 201. doi: 10.1134/S0081543807010105.

[21]

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

[22]

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

[23]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7.

[24]

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

[25]

J. V. Poncelet, Traité des Propriétés Projectives des Figures,, Mett, (1822).

[26]

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

[27]

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

[28]

S. Tabachnikov, Geometry and Billiards,, Student Mathematical Library, (2005).

[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.

[30]

M. Viana, Dynamics of Interval Exchange Maps and Teichmüller flows,, Lecture Notes, (2008).

[31]

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

[32]

A. Zorich, Flat surfaces,, in Frontiers in Number Theory, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

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