Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation

Serge Cantat[1]; Frank Loray[1]

  • [1] Université de Rennes 1 IRMAR - CNRS Campus de Beaulieu 35042 Rennes cedex (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 2927-2978
  • ISSN: 0373-0956

Abstract

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

How to cite

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Cantat, Serge, and Loray, Frank. "Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation." Annales de l’institut Fourier 59.7 (2009): 2927-2978. <http://eudml.org/doc/10476>.

@article{Cantat2009,
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.},
affiliation = {Université de Rennes 1 IRMAR - CNRS Campus de Beaulieu 35042 Rennes cedex (France); Université de Rennes 1 IRMAR - CNRS Campus de Beaulieu 35042 Rennes cedex (France)},
author = {Cantat, Serge, Loray, Frank},
journal = {Annales de l’institut Fourier},
keywords = {Painlevé equations; holomorphic foliations; character varieties; geometric structures; orbits; Malgrange irreducibility},
language = {eng},
number = {7},
pages = {2927-2978},
publisher = {Association des Annales de l’institut Fourier},
title = {Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation},
url = {http://eudml.org/doc/10476},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2927
EP - 2978
AB - 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.
LA - eng
KW - Painlevé equations; holomorphic foliations; character varieties; geometric structures; orbits; Malgrange irreducibility
UR - http://eudml.org/doc/10476
ER -

References

top
  1. Roger C. Alperin, An elementary account of Selberg’s lemma, Enseign. Math. (2) 33 (1987), 269-273 Zbl0639.20030MR925989
  2. Robert L. Benedetto, William M. Goldman, The topology of the relative character varieties of a quadruply-punctured sphere, Experiment. Math. 8 (1999), 85-103 Zbl0957.57003MR1685040
  3. Joan S. Birman, Braids, links, and mapping class groups, (1974), Princeton University Press, Princeton, N.J. Zbl0305.57013MR375281
  4. Philip Boalch, Towards a nonlinear Schwarz’s list, arXiv:0707.3375v1 [math.CA] (2007), 1-28 MR2322328
  5. J. W. Bruce, C. T. C. Wall, On the classification of cubic surfaces, J. London Math. Soc. (2) 19 (1979), 245-256 Zbl0393.14007MR533323
  6. Serge Cantat, Bers and Hénon, Painlevé and Schrödinger, Duke Math. J. (to appear), 1-41 Zbl1181.37068MR2553877
  7. Serge Cantat, Frank Loray, Holomorphic dynamics, Painlevé VI equation and character varieties, arXiv:0711.1579v2 [math.DS] (2007), 1-69 Zbl1204.34123
  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 (2007), 15-20, Res. Inst. Math. Sci. (RIMS), Kyoto Zbl1219.34113MR2310018
  9. Guy Casale, Le groupoïde de Galois de P 1 et son irréductibilité, Comment. Math. Helv. 83 (2008), 471-519 Zbl1163.34060MR2410777
  10. Guy Casale, Une preuve Galoisienne de l’irréductibilité au sens de Nishioka-Umemura de la première équation de Painlevé, Astérisque (2008), 83-100 
  11. Boris Dubrovin, Marta Mazzocco, Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147 Zbl0960.34075MR1767271
  12. Marat H. Èlʼ-Huti, Cubic surfaces of Markov type, Mat. Sb. (N.S.) 93(135) (1974), 331-346, 487 Zbl0293.14012MR342518
  13. William M. Goldman, Ergodic theory on moduli spaces, Ann. of Math. (2) 146 (1997), 475-507 Zbl0907.57009MR1491446
  14. William M. Goldman, The modular group action on real SL ( 2 ) -characters of a one-holed torus, Geom. Topol. 7 (2003), 443-486 (electronic) Zbl1037.57001MR2026539
  15. William M. Goldman, Mapping class group dynamics on surface group representations, Problems on mapping class groups and related topics 74 (2006), 189-214, Amer. Math. Soc., Providence, RI Zbl1304.57025MR2264541
  16. Robert D. Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972), 635-649 Zbl1184.20009MR314993
  17. Robert D. Horowitz, Induced automorphisms on Fricke characters of free groups, Trans. Amer. Math. Soc. 208 (1975), 41-50 Zbl0306.20027MR369540
  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), 1-30 Zbl1087.34062MR2036953
  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), 103-167 Zbl1161.34063MR2353464
  20. Nikolai V. Ivanov, Mapping class groups, Handbook of geometric topology (2002), 523-633, North-Holland, Amsterdam Zbl1002.57001MR1886678
  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 (2007), 143-156, Aoki, T.; Takei, Y.; Tose, N.; Majima, H. (Eds.) Zbl1177.34112
  22. Katsunori Iwasaki, Finite branch solutions to Painlevé VI around a fixed singular point, Adv. Math. 217 (2008), 1889-1934 Zbl1163.34061MR2388081
  23. Katsunori Iwasaki, Takato Uehara, An ergodic study of Painlevé VI, Math. Ann. 338 (2007), 295-345 Zbl1136.34067MR2302065
  24. Oleg Lisovyy, Yuriy Tykhyy, Algebraic solutions of the sixth Painlevé equation, arXiv:0809.4873v2 [math.CA] (2008), 1-53 Zbl1307.34135
  25. Bernard Malgrange, Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2 38 (2001), 465-501, Enseignement Math., Geneva Zbl1033.32020MR1929336
  26. Marta Mazzocco, Picard and Chazy solutions to the Painlevé VI equation, Math. Ann. 321 (2001), 157-195 Zbl0999.34079MR1857373
  27. David Mumford, Caroline Series, David Wright, Indra’s pearls, (2002), Cambridge University Press, New York Zbl1141.00002MR1913879
  28. Keiji Nishioka, A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J. 109 (1988), 63-67 Zbl0613.34030MR931951
  29. Masatoshi Noumi, Yasuhiko Yamada, A new Lax pair for the sixth Painlevé equation associated with s o ^ ( 8 ) , Microlocal analysis and complex Fourier analysis (2002), 238-252, World Sci. Publ., River Edge, NJ Zbl1047.34105MR2068540
  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), 1-79 Zbl0426.58017MR614694
  31. Kazuo Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation P VI , Ann. Mat. Pura Appl. (4) 146 (1987), 337-381 Zbl0637.34019MR916698
  32. Joseph P. Previte, Eugene Z. Xia, Exceptional discrete mapping class group orbits in moduli spaces, Forum Math. 15 (2003), 949-954 Zbl1036.57008MR2010288
  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), 65-72 Zbl1083.57026MR2163890
  34. Masa-Hiko Saito, Taro Takebe, Hitomi Terajima, Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom. 11 (2002), 311-362 Zbl1022.34079MR1874117
  35. Masa-Hiko Saito, Hitomi Terajima, Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ. 44 (2004), 529-568 Zbl1117.14015MR2103782
  36. Hiroshi Umemura, Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J. 117 (1990), 125-171 Zbl0688.34006MR1044939
  37. Humihiko Watanabe, Birational canonical transformations and classical solutions of the sixth Painlevé equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 379-425 (1999) Zbl0933.34095MR1678014

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