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, Tome 59 (2009) no. 7, pp. 2927-2978.

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.

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.

DOI : 10.5802/aif.2512
Classification : 34M55, 37F75, 20C15, 57M50
Keywords: Painlevé equations, holomorphic foliations, character varieties, geometric structures
Mot clés : équations de Painlevé, feuilletages holomorphes, variétés des caractères, structures géométriques
Cantat, Serge 1 ; Loray, Frank 1

1 Université de Rennes 1 IRMAR - CNRS Campus de Beaulieu 35042 Rennes cedex (France)
@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},
     mrnumber = {2649343},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2512/}
}
TY  - JOUR
AU  - Cantat, Serge
AU  - Loray, Frank
TI  - Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2927
EP  - 2978
VL  - 59
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://www.numdam.org/articles/10.5802/aif.2512/
DO  - 10.5802/aif.2512
LA  - en
ID  - AIF_2009__59_7_2927_0
ER  - 
%0 Journal Article
%A Cantat, Serge
%A Loray, Frank
%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://www.numdam.org/articles/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, Tome 59 (2009) no. 7, pp. 2927-2978. doi : 10.5802/aif.2512. https://www.numdam.org/articles/10.5802/aif.2512/

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

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

[3] Birman, Joan S. Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974 (Annals of Mathematics Studies, No. 82) | MR | Zbl

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

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

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

[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

[9] Casale, Guy Le groupoïde de Galois de P1 et son irréductibilité, Comment. Math. Helv., Volume 83 (2008) no. 3, pp. 471-519 | DOI | MR | Zbl

[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., Volume 141 (2000) no. 1, pp. 55-147 | DOI | MR | Zbl

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

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

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

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

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

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

[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., Volume 1 (2004), pp. 1-30 | DOI | MR | Zbl

[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 | Zbl

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

[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

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

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

[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.), Volume 38, Enseignement Math., Geneva, 2001, pp. 465-501 | MR | Zbl

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

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

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

[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 | Zbl

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

[31] Okamoto, Kazuo Studies on the Painlevé equations. I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. (4), Volume 146 (1987), pp. 337-381 | DOI | MR | Zbl

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

[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, Volume 112 (2005), pp. 65-72 | DOI | MR | Zbl

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

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

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

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

  • Klimeš, Martin Wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence, Annales de l'Institut Fourier, Volume 74 (2024) no. 1, p. 121 | DOI:10.5802/aif.3579
  • Fuchs, Elena; Litman, Matthew; Silverman, Joseph H.; Tran, Austin Orbits on K3 Surfaces of Markoff Type, Experimental Mathematics, Volume 33 (2024) no. 4, p. 663 | DOI:10.1080/10586458.2023.2239265
  • Paul, Emmanuel; Ramis, Jean-Pierre Dynamics of the Fifth Painlevé Foliation, Handbook of Geometry and Topology of Singularities VI: Foliations (2024), p. 307 | DOI:10.1007/978-3-031-54172-8_9
  • Ramis, Jean-Pierre Epilogue: Stokes Phenomena. Dynamics, Classification Problems and Avatars, Handbook of Geometry and Topology of Singularities VI: Foliations (2024), p. 383 | DOI:10.1007/978-3-031-54172-8_10
  • Dao, Quang-Duc Brauer–Manin obstruction for Wehler K3 surfaces of Markoff type, International Journal of Number Theory, Volume 20 (2024) no. 08, p. 1967 | DOI:10.1142/s1793042124500969
  • Dao, Quang-Duc Brauer–Manin obstruction for integral points on Markoff-type cubic surfaces, Journal of Number Theory, Volume 254 (2024), p. 65 | DOI:10.1016/j.jnt.2023.07.007
  • Hrushovski, Ehud Approximate equivalence relations, Model Theory, Volume 3 (2024) no. 2, p. 317 | DOI:10.2140/mt.2024.3.317
  • Bousseau, Pierrick Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus, Communications in Mathematical Physics, Volume 398 (2023) no. 1, p. 1 | DOI:10.1007/s00220-022-04512-9
  • Goldman, William Compact components of planar surface group representations, Computational Aspects of Discrete Subgroups of Lie Groups, Volume 783 (2023), p. 69 | DOI:10.1090/conm/783/15701
  • Ballandras, Mathieu Intersection cohomology of character varieties for punctured Riemann surfaces, Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), p. 141 | DOI:10.5802/jep.215
  • Blázquez-Sanz, David; Casale, Guy; Arboleda, Juan Sebastián Díaz The Malgrange–Galois groupoid of the Painlevé VI equation with parameters, Advances in Geometry, Volume 22 (2022) no. 3, p. 301 | DOI:10.1515/advgeom-2022-0010
  • Casale, Guy; Davy, Damien Spécialisation du groupoïde de Galois d’un champ de vecteurs, Annales de l'Institut Fourier, Volume 72 (2022) no. 6, p. 2399 | DOI:10.5802/aif.3506
  • Biswas, Indranil; Gupta, Subhojoy; Mj, Mahan; Whang, Junho Peter Surface group representations in SL2(ℂ) with finite mapping class orbits, Geometry Topology, Volume 26 (2022) no. 2, p. 679 | DOI:10.2140/gt.2022.26.679
  • Ghosh, Amit; Sarnak, Peter Integral points on Markoff type cubic surfaces, Inventiones mathematicae, Volume 229 (2022) no. 2, p. 689 | DOI:10.1007/s00222-022-01114-z
  • de Courcy-Ireland, Matthew; Magee, Michael Kesten–McKay law for the Markoff surface mod p, Annales Henri Lebesgue, Volume 4 (2021), p. 227 | DOI:10.5802/ahl.71
  • Ramis, Jean-Pierre Hiroshi Umemura et les mathématiques françaises, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 29 (2021) no. 5, p. 1007 | DOI:10.5802/afst.1655
  • Ohyama, Yousuke; Ramis, Jean-Pierre; Sauloy, Jacques The space of monodromy data for the Jimbo–Sakai family ofq-difference equations, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 29 (2021) no. 5, p. 1119 | DOI:10.5802/afst.1659
  • Cantat, Serge Endomorphisms and bijections of the character variety χ(F 2 ,SL 2 (C)), Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 29 (2020) no. 4, p. 897 | DOI:10.5802/afst.1648
  • Diarra, Karamoko; Loray, Frank Classification of algebraic solutions of irregular Garnier systems, Compositio Mathematica, Volume 156 (2020) no. 5, p. 881 | DOI:10.1112/s0010437x20007083
  • Gavrylenko, Pavlo; Santachiara, Raoul Crossing invariant correlation functions at c = 1 from isomonodromic τ functions, Journal of High Energy Physics, Volume 2019 (2019) no. 11 | DOI:10.1007/jhep11(2019)119
  • Goldman, William; McShane, Greg; Stantchev, George; Tan, Ser Peow Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane, Memoirs of the American Mathematical Society, Volume 259 (2019) no. 1249 | DOI:10.1090/memo/1249
  • Buchstaber, V. M.; Veselov, A. P. Conway topograph, -dynamics and two-valued groups, Russian Mathematical Surveys, Volume 74 (2019) no. 3, p. 387 | DOI:10.1070/rm9886
  • Buchstaber, Victor Matveevich; Veselov, Aleksandr Petrovich Топограф Конвея, PGL2(Z)-динамика и двузначные группы, Успехи математических наук, Volume 74 (2019) no. 3(447), p. 17 | DOI:10.4213/rm9886
  • Cousin, Gaël; Moussard, Delphine Finite Braid group orbits in Aff(C)-character varieties of the punctured sphere, International Mathematics Research Notices, Volume 2018 (2018) no. 11, p. 3388 | DOI:10.1093/imrn/rnw283
  • Casale, Guy; Weil, Jacques-Arthur Galoisian methods for testing irreducibility of order two nonlinear differential equations, Pacific Journal of Mathematics, Volume 297 (2018) no. 2, p. 299 | DOI:10.2140/pjm.2018.297.299
  • Cousin, Gaël Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties, Mathematische Annalen, Volume 367 (2017) no. 3-4, p. 965 | DOI:10.1007/s00208-016-1397-y
  • Spalding, K; Veselov, A P Lyapunov spectrum of Markov and Euclid trees, Nonlinearity, Volume 30 (2017) no. 12, p. 4428 | DOI:10.1088/1361-6544/aa88ff
  • Bourgain, Jean; Gamburd, Alexander; Sarnak, Peter Markoff triples and strong approximation, Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, p. 131 | DOI:10.1016/j.crma.2015.12.006
  • Chekhov, Leonid O.; Mazzocco, Marta; Rubtsov, Vladimir N. Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras, International Mathematics Research Notices (2016), p. rnw219 | DOI:10.1093/imrn/rnw219
  • Loray, F Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry, Izvestiya: Mathematics, Volume 80 (2016) no. 1, p. 113 | DOI:10.1070/im8310
  • Loray, Frank Изомонодромные деформации связностей Ламе, уравнение Пенлеве VI и симметрия Окамото, Известия Российской академии наук. Серия математическая, Volume 80 (2016) no. 1, p. 119 | DOI:10.4213/im8310
  • Girand, Arnaud Dynamical Green functions and discrete Schrödinger operators with potentials generated by primitive invertible substitution, Nonlinearity, Volume 27 (2014) no. 3, p. 527 | DOI:10.1088/0951-7715/27/3/527
  • Ben Hamed, Bassem; Gavrilov, Lubomir; Klughertz, Martine The holonomy group at infinity of the Painlevé VI equation, Journal of Mathematical Physics, Volume 53 (2012) no. 2 | DOI:10.1063/1.3681897
  • BLANC, JÉRÉMY; DÉSERTI, JULIE Embeddings of SL(2; ℤ) into the cremona group, Transformation Groups, Volume 17 (2012) no. 1, p. 21 | DOI:10.1007/s00031-012-9174-9

Cité par 34 documents. Sources : Crossref