We prove that the holonomy group at infinity of the Painlevé VI equation is virtually commutative.
REFERENCES
1.
Cantat
, Loray F.
, , “Holomorphic dynamics, Painlevé VI equation, and character varieties
,” Ann. Inst. Fourier
59
(7
), 2927
–2978
(2009
).2.
Casale
, G.
, “Une preuve galoisienne de l'irréductibilité au sens de Nishioka-Umemura de la 1ere équation de Painlevé
,” Astérisque
323
, 83
–100
(2009
).3.
Casale
, G.
, “The Galois groupoid of Picard-Painlevé sixth equation
,” RIMS Kôkyûroku Bessatsu, Res. Inst. Math. Sci. (RIMS), Kyoto
, B2
, 15
–20
(2007
).4.
Casale
, G.
, “An introduction to Malgrange pseudogroup
,” Séminaires et Congrès
23
, 89
–113
(2011
).5.
Casale
, G.
, “Liouvillian first integrals of differential equations
,” Banach Center Publ.
94
, 153
–161
(2011
).6.
Hain
, R. M.
, “The geometry of the mixed Hodge structure on the fundamental group
,” Proc. Symp. Pure Math.
46
, 247
–281
(1987
).7.
Horozov
, E.
and Stoyanova
, T.
, “Non-Integrability of some Painlevé VI equations and dilogarithms
,” Regular Chaotic Dyn.
12
(6
), 622
–629
(2007
).8.
Iwasaki
, K.
, Kimura
, H.
, Shimomura
, S.
, and Yoshida
, M.
, “From Gauss to Painlevé, A Modern Theory of Special Functions
,” Aspects of Mathematics
, E16 (Vieweg & Shon
, Friedr. Vieweg & Sohn, Braunschweig
, 1991
).9.
Malgrange
, B.
, “Le groupoide de Galois d'un feuilletage
,” Monographie 38, Enseign. Math.
2
, 465
–501
(2001
).10.
Morales-Ruiz
, J. J.
and Ramis
, J. P.
, “Galoisian obstructions to integrability of Hamiltonian systems. I, II
,” Methods Appl. Anal.
8
(1
), 33
–95
, 97–111, (2001
).11.
Morales-Ruiz
, J. J.
, Ramis
, J. P.
, and Simo
, C.
, “Integrability of Hamiltonian systems and differential Galois groups of higher variational equations
,” Ann. Sci. École Norm. Sup.
40
(6
), 845
–884
(2007
).12.
Morales-Ruiz
, J. J.
, “A remark about the Painlevé transcendents
,” Sémin. Congr.
14
, 229
–235
(2006
).13.
Morales-Ruiz
, J. J.
, “Differential Galois theory and non-integrability of Hamiltonian systems
,” Prog. Math.
179
, Birkhäuser Verlag, Basel, 1999
.14.
Morales-Ruiz
, J. J.
, Ramis
, J.
, “Pierre Integrability of dynamical systems through differential Galois theory: a practical guide
,” in Differential Algebra, Complex Analysis and Orthogonal Polynomials
(American Mathematics Society
, Providence, RI
, 2010
), pp. 143
–220
, 509.15.
Okamoto
, K.
, “Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé
,” Japan. J. Math. (N.S.)
5
(1
), 1
–79
(1979
).16.
Stoyanova
, T.
, “Non-integrability of Painlevé VI equations in the Liouville sense
,” Nonlinearity
22
(9
), 2201
–2230
(2009
).17.
Tretkoff
, C.
and Tretkoff
, M.
, “Solution of the inverse problem of differential Galois theory in the classical case
,” Am. J. Math.
101
(6
), 1327
–1332
(1979
).18.
Watanabe
, H.
, “Birational canonical transformations and classical solutions of the sixth Painlevé equation
,” Ann. Scuola Norm. Sup. Pisa Cl. Sci.
(4
)27
, (1998), no. 3-4, 379
–425
(1999
).© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.