logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Serge Cantat; Frank Loray
Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation
(Dynamique sur la variété des caractères et irréductibilité au sens de Malgrange de l’équation de Painlevé VI)
Annales de l'institut Fourier, 59 no. 7 (2009), p. 2927-2978, doi: 10.5802/aif.2512
Article PDF | Reviews MR 2649343 | Zbl pre05689411
Class. Math.: 34M55, 37F75, 20C15, 57M50
Keywords: Painlevé equations, holomorphic foliations, character varieties, geometric structures

Résumé - Abstract

We consider representations of the fundamental group of the four punctured sphere into $\mathrm{SL}(2,\mathbb{C})$. The moduli space of representations modulo conjugacy is the character variety. The Mapping Class Group of the punctured sphere acts on this space by symplectic polynomial automorphisms. This dynamical system can be interpreted as the monodromy of the Painlevé VI equation. Infinite bounded orbits are characterized: they come from $\sf {SU}(2)$-representations. We prove the absence of invariant affine structure (and invariant foliation) for this dynamical system except for special explicit parameters. Following results of Casale, this implies that Malgrange’s groupoid of the Painlevé VI foliation coincides with the symplectic one. This provides a new proof of the transcendence of Painlevé solutions.

Bibliography

[1] Roger C. Alperin, “An elementary account of Selberg’s lemma”, Enseign. Math. (2) 33 (1987) no. 3-4, p. 269-273  MR 925989 |  Zbl 0639.20030
[2] Robert L. Benedetto & William M. Goldman, “The topology of the relative character varieties of a quadruply-punctured sphere”, Experiment. Math. 8 (1999) no. 1, p. 85-103 Article |  MR 1685040 |  Zbl 0957.57003
[3] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974, Annals of Mathematics Studies, No. 82  MR 375281 |  Zbl 0305.57013
[4] Philip Boalch, “Towards a nonlinear Schwarz’s list”, arXiv:0707.3375v1 [math.CA] (2007), p. 1-28 arXiv |  MR 2322328
[5] J. W. Bruce & C. T. C. Wall, “On the classification of cubic surfaces”, J. London Math. Soc. (2) 19 (1979) no. 2, p. 245-256 Article |  MR 533323 |  Zbl 0393.14007
[6] Serge Cantat, “Bers and Hénon, Painlevé and Schrödinger”, Duke Math. J. (to appear), p. 1-41  MR 2553877 |  Zbl pre05611495
[7] Serge Cantat & Frank Loray, “Holomorphic dynamics, Painlevé VI equation and character varieties”, arXiv:0711.1579v2 [math.DS] (2007), p. 1-69 arXiv
[8] Guy Casale, The Galois groupoid of Picard-Painlevé VI equation, Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, p. 15–20  MR 2310018 |  Zbl pre05152679
[9] Guy Casale, “Le groupoïde de Galois de $P_1$ et son irréductibilité”, Comment. Math. Helv. 83 (2008) no. 3, p. 471-519 Article |  MR 2410777 |  Zbl 1163.34060
[10] Guy Casale, “Une preuve Galoisienne de l’irréductibilité au sens de Nishioka-Umemura de la première équation de Painlevé”, Astérisque (2008) no. 157, p. 83-100, Équations différentielles et singularités, en l’honneur de J. M. Aroca
[11] Boris Dubrovin & Marta Mazzocco, “Monodromy of certain Painlevé-VI transcendents and reflection groups”, Invent. Math. 141 (2000) no. 1, p. 55-147 Article |  MR 1767271 |  Zbl 0960.34075
[12] Marat H. Èlʼ-Huti, “Cubic surfaces of Markov type”, Mat. Sb. (N.S.) 93(135) (1974), p. 331-346, 487  MR 342518 |  Zbl 0293.14012
[13] William M. Goldman, “Ergodic theory on moduli spaces”, Ann. of Math. (2) 146 (1997) no. 3, p. 475-507 Article |  MR 1491446 |  Zbl 0907.57009
[14] William M. Goldman, “The modular group action on real ${\rm SL}(2)$-characters of a one-holed torus”, Geom. Topol. 7 (2003), p. 443-486 (electronic) Article |  MR 2026539 |  Zbl 1037.57001
[15] William M. Goldman, Mapping class group dynamics on surface group representations, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., 2006, p. 189–214  MR 2264541 |  Zbl pre05124684
[16] Robert D. Horowitz, “Characters of free groups represented in the two-dimensional special linear group”, Comm. Pure Appl. Math. 25 (1972), p. 635-649 Article |  MR 314993
[17] Robert D. Horowitz, “Induced automorphisms on Fricke characters of free groups”, Trans. Amer. Math. Soc. 208 (1975), p. 41-50 Article |  MR 369540 |  Zbl 0306.20027
[18] Michi-aki Inaba, Katsunori Iwasaki & Masa-Hiko Saito, “Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence”, Int. Math. Res. Not. 1 (2004), p. 1-30 Article |  MR 2036953 |  Zbl 1087.34062
[19] Michi-aki Inaba, Katsunori Iwasaki & Masa-Hiko Saito, “Dynamics of the sixth Painlevé equation, in Théories asymptotiques et équations de Painlevé”, Séminaires et Congrès (2006) no. 14, p. 103-167  MR 2353464 |  Zbl 1161.34063
[20] Nikolai V. Ivanov, Mapping class groups, Handbook of geometric topology, North-Holland, 2002, p. 523–633  MR 1886678 |  Zbl 1002.57001
[21] Katsunori Iwasaki, Some dynamical aspects of Painlevé VI, Algebraic Analysis of Differential Equations, In honor of Prof. Takahiro KAWAI on the occasion of his sixtieth birthday, Aoki, T.; Takei, Y.; Tose, N.; Majima, H. (Eds.), 2007, p. 143–156  Zbl pre05258283
[22] Katsunori Iwasaki, “Finite branch solutions to Painlevé VI around a fixed singular point”, Adv. Math. 217 (2008) no. 5, p. 1889-1934 Article |  MR 2388081 |  Zbl 1163.34061
[23] Katsunori Iwasaki & Takato Uehara, “An ergodic study of Painlevé VI”, Math. Ann. 338 (2007) no. 2, p. 295-345 Article |  MR 2302065 |  Zbl 1136.34067
[24] Oleg Lisovyy & Yuriy Tykhyy, “Algebraic solutions of the sixth Painlevé equation”, arXiv:0809.4873v2 [math.CA] (2008), p. 1-53 arXiv
[25] Bernard Malgrange, Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math. 38, Enseignement Math., 2001, p. 465–501  MR 1929336 |  Zbl 1033.32020
[26] Marta Mazzocco, “Picard and Chazy solutions to the Painlevé VI equation”, Math. Ann. 321 (2001) no. 1, p. 157-195 Article |  MR 1857373 |  Zbl 0999.34079
[27] David Mumford, Caroline Series & David Wright, Indra’s pearls, Cambridge University Press, New York, 2002, The vision of Felix Klein  MR 1913879 |  Zbl 1141.00002
[28] Keiji Nishioka, “A note on the transcendency of Painlevé’s first transcendent”, Nagoya Math. J. 109 (1988), p. 63-67 Article |  MR 931951 |  Zbl 0613.34030
[29] Masatoshi Noumi & Yasuhiko Yamada, A new Lax pair for the sixth Painlevé equation associated with $\widehat{{so}}(8)$, Microlocal analysis and complex Fourier analysis, World Sci. Publ., River Edge, NJ, 2002, p. 238–252  MR 2068540 |  Zbl 1047.34105
[30] Kazuo Okamoto, “Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé”, Japan. J. Math. (N.S.) 5 (1979) no. 1, p. 1-79  MR 614694 |  Zbl 0426.58017
[31] Kazuo Okamoto, “Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{{\rm VI}}$”, Ann. Mat. Pura Appl. (4) 146 (1987), p. 337-381 Article |  MR 916698 |  Zbl 0637.34019
[32] Joseph P. Previte & Eugene Z. Xia, “Exceptional discrete mapping class group orbits in moduli spaces”, Forum Math. 15 (2003) no. 6, p. 949-954 Article |  MR 2010288 |  Zbl 1036.57008
[33] Joseph P. Previte & Eugene Z. Xia, “Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy”, Geom. Dedicata 112 (2005), p. 65-72 Article |  MR 2163890 |  Zbl 1083.57026
[34] Masa-Hiko Saito, Taro Takebe & Hitomi Terajima, “Deformation of Okamoto-Painlevé pairs and Painlevé equations”, J. Algebraic Geom. 11 (2002) no. 2, p. 311-362 Article |  MR 1874117 |  Zbl 1022.34079
[35] Masa-Hiko Saito & Hitomi Terajima, “Nodal curves and Riccati solutions of Painlevé equations”, J. Math. Kyoto Univ. 44 (2004) no. 3, p. 529-568 Article |  MR 2103782 |  Zbl 1117.14015
[36] Hiroshi Umemura, “Second proof of the irreducibility of the first differential equation of Painlevé”, Nagoya Math. J. 117 (1990), p. 125-171 Article |  MR 1044939 |  Zbl 0688.34006
[37] Humihiko Watanabe, “Birational canonical transformations and classical solutions of the sixth Painlevé equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998) no. 3-4, p. 379-425 (1999) Numdam |  MR 1678014 |  Zbl 0933.34095
top