Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2927-2978.

We consider representations of the fundamental group of the four punctured sphere into SL(2,). 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 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.

Nous étudions l’action du groupe modulaire sur l’espace des représentations du groupe fondamental de la sphère privée de quatre points dans SL(2,). Ce système dynamique peut être interprété comme la monodromie de l’équation de Painlevé VI. Nous caractérisons les orbites infinies bornées : elles proviennent des représentations dans SU(2). Nous démontrons l’absence de struture affine invariante (excepté pour des paramètres spéciaux) puis déduisons, en nous appuyant sur des travaux de Casale, que le groupoïde de Malgrange associé est le groupoïde symplectique. Ceci permet de donner une preuve de l’irréductibilité de l’équation de Painlevé VI, c’est-à-dire de la forte transcendance de ses solutions, par une approche galoisienne, dans l’esprit de la tentative de Drach et Painlevé.

Received:
Accepted:
DOI: 10.5802/aif.2512
Classification: 34M55,  37F75,  20C15,  57M50
Keywords: Painlevé equations, holomorphic foliations, character varieties, geometric structures
@article{AIF_2009__59_7_2927_0,
     author = {Cantat, Serge and Loray, Frank},
     title = {Dynamics on {Character} {Varieties} and {Malgrange} irreducibility of {Painlev\'e} {VI} equation},
     journal = {Annales de l'Institut Fourier},
     pages = {2927--2978},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     doi = {10.5802/aif.2512},
     zbl = {pre05689411},
     mrnumber = {2649343},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2512/}
}
TY  - JOUR
TI  - Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation
JO  - Annales de l'Institut Fourier
PY  - 2009
DA  - 2009///
SP  - 2927
EP  - 2978
VL  - 59
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2512/
UR  - https://zbmath.org/?q=an%3Apre05689411
UR  - https://www.ams.org/mathscinet-getitem?mr=2649343
UR  - https://doi.org/10.5802/aif.2512
DO  - 10.5802/aif.2512
LA  - en
ID  - AIF_2009__59_7_2927_0
ER  - 
%0 Journal Article
%T Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation
%J Annales de l'Institut Fourier
%D 2009
%P 2927-2978
%V 59
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2512
%R 10.5802/aif.2512
%G en
%F AIF_2009__59_7_2927_0
Cantat, Serge; Loray, Frank. Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2927-2978. doi : 10.5802/aif.2512. https://aif.centre-mersenne.org/articles/10.5802/aif.2512/

[1] Alperin, Roger C. An elementary account of Selberg’s lemma, Enseign. Math. (2), Tome 33 (1987) no. 3-4, pp. 269-273 | MR: 925989 | Zbl: 0639.20030

[2] Benedetto, Robert L.; Goldman, William M. The topology of the relative character varieties of a quadruply-punctured sphere, Experiment. Math., Tome 8 (1999) no. 1, pp. 85-103 | MR: 1685040 | Zbl: 0957.57003

[3] Birman, Joan S. 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] Boalch, Philip Towards a nonlinear Schwarz’s list, arXiv:0707.3375v1 [math.CA] (2007), pp. 1-28 | MR: 2322328

[5] Bruce, J. W.; Wall, C. T. C. On the classification of cubic surfaces, J. London Math. Soc. (2), Tome 19 (1979) no. 2, pp. 245-256 | Article | MR: 533323 | Zbl: 0393.14007

[6] Cantat, Serge Bers and Hénon, Painlevé and Schrödinger, Duke Math. J. (to appear), pp. 1-41 | MR: 2553877 | Zbl: pre05611495

[7] Cantat, Serge; Loray, Frank Holomorphic dynamics, Painlevé VI equation and character varieties, arXiv:0711.1579v2 [math.DS] (2007), pp. 1-69

[8] Casale, Guy 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, pp. 15-20 | MR: 2310018 | Zbl: pre05152679

[9] Casale, Guy Le groupoïde de Galois de P 1 et son irréductibilité, Comment. Math. Helv., Tome 83 (2008) no. 3, pp. 471-519 | Article | MR: 2410777 | Zbl: 1163.34060

[10] Casale, Guy Une preuve Galoisienne de l’irréductibilité au sens de Nishioka-Umemura de la première équation de Painlevé, Astérisque (2008) no. 157, pp. 83-100 (Équations différentielles et singularités, en l’honneur de J. M. Aroca)

[11] Dubrovin, Boris; Mazzocco, Marta Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math., Tome 141 (2000) no. 1, pp. 55-147 | Article | MR: 1767271 | Zbl: 0960.34075

[12] Èlʼ-Huti, Marat H. Cubic surfaces of Markov type, Mat. Sb. (N.S.), Tome 93(135) (1974), p. 331-346, 487 | MR: 342518 | Zbl: 0293.14012

[13] Goldman, William M. Ergodic theory on moduli spaces, Ann. of Math. (2), Tome 146 (1997) no. 3, pp. 475-507 | Article | MR: 1491446 | Zbl: 0907.57009

[14] Goldman, William M. The modular group action on real SL (2)-characters of a one-holed torus, Geom. Topol., Tome 7 (2003), p. 443-486 (electronic) | Article | MR: 2026539 | Zbl: 1037.57001

[15] Goldman, William M. Mapping class group dynamics on surface group representations, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.) Tome 74, Amer. Math. Soc., Providence, RI, 2006, pp. 189-214 | MR: 2264541 | Zbl: pre05124684

[16] Horowitz, Robert D. Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math., Tome 25 (1972), pp. 635-649 | Article | MR: 314993

[17] Horowitz, Robert D. Induced automorphisms on Fricke characters of free groups, Trans. Amer. Math. Soc., Tome 208 (1975), pp. 41-50 | Article | MR: 369540 | Zbl: 0306.20027

[18] Inaba, Michi-aki; Iwasaki, Katsunori; Saito, Masa-Hiko Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence, Int. Math. Res. Not., Tome 1 (2004), pp. 1-30 | Article | MR: 2036953 | Zbl: 1087.34062

[19] Inaba, Michi-aki; Iwasaki, Katsunori; Saito, Masa-Hiko Dynamics of the sixth Painlevé equation, in Théories asymptotiques et équations de Painlevé, Séminaires et Congrès (2006) no. 14, pp. 103-167 | MR: 2353464 | Zbl: 1161.34063

[20] Ivanov, Nikolai V. Mapping class groups, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 523-633 | MR: 1886678 | Zbl: 1002.57001

[21] Iwasaki, Katsunori 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, pp. 143-156 | Zbl: pre05258283

[22] Iwasaki, Katsunori Finite branch solutions to Painlevé VI around a fixed singular point, Adv. Math., Tome 217 (2008) no. 5, pp. 1889-1934 | Article | MR: 2388081 | Zbl: 1163.34061

[23] Iwasaki, Katsunori; Uehara, Takato An ergodic study of Painlevé VI, Math. Ann., Tome 338 (2007) no. 2, pp. 295-345 | Article | MR: 2302065 | Zbl: 1136.34067

[24] Lisovyy, Oleg; Tykhyy, Yuriy Algebraic solutions of the sixth Painlevé equation, arXiv:0809.4873v2 [math.CA] (2008), pp. 1-53

[25] Malgrange, Bernard Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2 (Monogr. Enseign. Math.) Tome 38, Enseignement Math., Geneva, 2001, pp. 465-501 | MR: 1929336 | Zbl: 1033.32020

[26] Mazzocco, Marta Picard and Chazy solutions to the Painlevé VI equation, Math. Ann., Tome 321 (2001) no. 1, pp. 157-195 | Article | MR: 1857373 | Zbl: 0999.34079

[27] Mumford, David; Series, Caroline; Wright, David Indra’s pearls, Cambridge University Press, New York, 2002 (The vision of Felix Klein) | MR: 1913879 | Zbl: 1141.00002

[28] Nishioka, Keiji A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J., Tome 109 (1988), pp. 63-67 | MR: 931951 | Zbl: 0613.34030

[29] Noumi, Masatoshi; Yamada, Yasuhiko A new Lax pair for the sixth Painlevé equation associated with so ^(8), Microlocal analysis and complex Fourier analysis, World Sci. Publ., River Edge, NJ, 2002, pp. 238-252 | MR: 2068540 | Zbl: 1047.34105

[30] Okamoto, Kazuo Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.), Tome 5 (1979) no. 1, pp. 1-79 | MR: 614694 | Zbl: 0426.58017

[31] Okamoto, Kazuo Studies on the Painlevé equations. I. Sixth Painlevé equation P VI , Ann. Mat. Pura Appl. (4), Tome 146 (1987), pp. 337-381 | Article | MR: 916698 | Zbl: 0637.34019

[32] Previte, Joseph P.; Xia, Eugene Z. Exceptional discrete mapping class group orbits in moduli spaces, Forum Math., Tome 15 (2003) no. 6, pp. 949-954 | Article | MR: 2010288 | Zbl: 1036.57008

[33] Previte, Joseph P.; Xia, Eugene Z. Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy, Geom. Dedicata, Tome 112 (2005), pp. 65-72 | Article | MR: 2163890 | Zbl: 1083.57026

[34] Saito, Masa-Hiko; Takebe, Taro; Terajima, Hitomi Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom., Tome 11 (2002) no. 2, pp. 311-362 | Article | MR: 1874117 | Zbl: 1022.34079

[35] Saito, Masa-Hiko; Terajima, Hitomi Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ., Tome 44 (2004) no. 3, pp. 529-568 | MR: 2103782 | Zbl: 1117.14015

[36] Umemura, Hiroshi Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J., Tome 117 (1990), pp. 125-171 | MR: 1044939 | Zbl: 0688.34006

[37] Watanabe, Humihiko Birational canonical transformations and classical solutions of the sixth Painlevé equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Tome 27 (1998) no. 3-4, p. 379-425 (1999) | Numdam | MR: 1678014 | Zbl: 0933.34095

Cited by Sources: