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Minimal yet measurable foliations
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 |
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. |
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