Pseudo-integrable billiards and arithmetic dynamics

Previous Article
Erratum: Billiards in nearly isosceles triangles
JMD Home
This Issue
Next Article
Minimal yet measurable foliations

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

Abstract
Related Papers

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.
[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.
[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.
[4]	H. J. M. Bos, C. Kers, F. Oort and D. W. Raven, Poncelet's closure theorem, Expo. Math., 5 (1987), 289-364.
[5]	A. Cayley, Note on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 6 (1853), 99-102.
[6]	A. Cayley, Developments on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 7 (1854), 339-345.
[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), 92-94.
[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.
[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.
[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.
[12]	V. Dragović and M. Radnović, Bifurcations of Liouville tori in elliptical billiards, Regul. Chaotic Dyn., 14 (2009), 479-494. 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, Birkhäuser/Springer Basel AG, Basel, 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), vi+123 pp. doi: 10.1090/memo/0464.
[15]	L. Flatto, Poncelet's Theorem, Chapter 15 by S. Tabachnikov, American Mathematical Society, Providence, RI, 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), 609-628. doi: 10.2307/2974690.
[19]	V. Kozlov and D. Treshchëv, Billiards, Amer. Math. Soc., Providence, RI, 1991.
[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.
[21]	M. Levi and S. Tabachnikov, The Poncelet grid and billiards in ellipses, Amer. Math. Monthly, 114 (2007), 895-908.
[22]	A. G. Maier, Trajectories on closable orientable surfaces, (Russian) Rec. Math. [Mat. Sbornik] N.S., 12(54) (1943), 71-84.
[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.
[24]	S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.
[25]	J. V. Poncelet, Traité des Propriétés Projectives des Figures, Mett, Paris, 1822.
[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.
[27]	R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175. doi: 10.1515/ADVGEOM.2007.010.
[28]	S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 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), 1-33.
[30]	M. Viana, Dynamics of Interval Exchange Maps and Teichmüller flows, Lecture Notes, 2008. Available from: http://w3.impa.br/~viana/out/ietf.pdf.
[31]	A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300.
[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.

show all references

References:

[1]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer Verlag, New York-Heidelberg, 1978.
[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.
[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.
[4]	H. J. M. Bos, C. Kers, F. Oort and D. W. Raven, Poncelet's closure theorem, Expo. Math., 5 (1987), 289-364.
[5]	A. Cayley, Note on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 6 (1853), 99-102.
[6]	A. Cayley, Developments on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 7 (1854), 339-345.
[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), 92-94.
[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.
[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.
[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.
[12]	V. Dragović and M. Radnović, Bifurcations of Liouville tori in elliptical billiards, Regul. Chaotic Dyn., 14 (2009), 479-494. 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, Birkhäuser/Springer Basel AG, Basel, 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), vi+123 pp. doi: 10.1090/memo/0464.
[15]	L. Flatto, Poncelet's Theorem, Chapter 15 by S. Tabachnikov, American Mathematical Society, Providence, RI, 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), 609-628. doi: 10.2307/2974690.
[19]	V. Kozlov and D. Treshchëv, Billiards, Amer. Math. Soc., Providence, RI, 1991.
[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.
[21]	M. Levi and S. Tabachnikov, The Poncelet grid and billiards in ellipses, Amer. Math. Monthly, 114 (2007), 895-908.
[22]	A. G. Maier, Trajectories on closable orientable surfaces, (Russian) Rec. Math. [Mat. Sbornik] N.S., 12(54) (1943), 71-84.
[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.
[24]	S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, (Russian) Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.
[25]	J. V. Poncelet, Traité des Propriétés Projectives des Figures, Mett, Paris, 1822.
[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.
[27]	R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175. doi: 10.1515/ADVGEOM.2007.010.
[28]	S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 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), 1-33.
[30]	M. Viana, Dynamics of Interval Exchange Maps and Teichmüller flows, Lecture Notes, 2008. Available from: http://w3.impa.br/~viana/out/ietf.pdf.
[31]	A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300.
[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.

[1]	Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123
[2]	W. Patrick Hooper. Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons. Journal of Modern Dynamics, 2007, 1 (4) : 649-663. doi: 10.3934/jmd.2007.1.649
[3]	Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[4]	Alexey Glutsyuk, Yury Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories. Journal of Modern Dynamics, 2012, 6 (3) : 287-326. doi: 10.3934/jmd.2012.6.287
[5]	Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[6]	Eugene Gutkin. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 367-382. doi: 10.3934/dcds.2006.16.367
[7]	Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[8]	Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[9]	Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006
[10]	Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[11]	Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253
[12]	Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[13]	Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[14]	Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077
[15]	Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[16]	Vladimir P. Burskii, Alexei S. Zhedanov. On Dirichlet, Poncelet and Abel problems. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1587-1633. doi: 10.3934/cpaa.2013.12.1587
[17]	Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022042
[18]	Jochen Brüning, Franz W. Kamber, Ken Richardson. The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations. Electronic Research Announcements, 2010, 17: 138-154. doi: 10.3934/era.2010.17.138
[19]	Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1
[20]	R. Bartolo, Anna Maria Candela, J.L. Flores, Addolorata Salvatore. Periodic trajectories in plane wave type spacetimes. Conference Publications, 2005, 2005 (Special) : 77-83. doi: 10.3934/proc.2005.2005.77

2020 Impact Factor: 0.848

Tools

Metrics

Other articles
by authors

on AIMS
Vladimir Dragović Milena Radnović
on Google Scholar
Vladimir Dragović Milena Radnović

[Back to Top]