On a general difference Galois theory. II. (English) Zbl 1194.12006

The authors apply the general Galois theory of difference equations introduced in the first part [see S. Morikawa, Ann. Inst. Fourier 59, No. 7, 2709–2732 (2009; Zbl 1194.12005)] to concrete examples. They study in detail discrete dynamical systems \((X,\phi)\) of iteration of a rational map \(\phi: X \to X\) on an algebraic curve \(X\) defined over a field \(C\) of characteristic 0 and so, in particular, on a compact Riemann surface \(X\) if \(C=\mathbb{C}\). In the last case they determine all the dynamical systems \((X,\phi)\) over an algebraic curve \(X\) such that the Lie algebra of their Galois group is finite-dimensional. For these dynamical systems, the Lie algebra is not only finite-dimensional but also solvable. So the authors call them infinitesimally solvable and yield their classification.


12H10 Difference algebra
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
58H05 Pseudogroups and differentiable groupoids
14H70 Relationships between algebraic curves and integrable systems


Zbl 1194.12005
Full Text: DOI Numdam EuDML


[1] Casale, G., Sur le groupoïde de Galois d’un feuilletage (2004)
[2] Casale, G., Enveloppe galoisienne d’une application rationnelle de \(\mathbb{P}^1\), Publ. Mat., 50, 191-202 (2006) · Zbl 1137.37022
[3] Demazure, M., Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup., (4) 3, 507-588 (1970) · Zbl 0223.14009
[4] Grothendieck, A., Séminaire Bourbaki, Vol. 6, Exp. No. 212, 99-118 (1995) · Zbl 0235.14007
[5] Kolchin, E., Differential algebra and algebraic groups, 54 (1973) · Zbl 0264.12102
[6] Lie, S., Théorie des Transformationsgruppen (1930)
[7] Malgrange, B., Essays on geometry and related topics, Vol. 1, 2, 38, 465-501 (2001) · Zbl 1033.32020
[8] Morikawa, S., On a general Galois theory of difference equations I, Ann. Inst. Fourier, Grenoble (2010)
[9] Silverman, J. H., The arithmetic of dynamical systems, 241 (2007) · Zbl 1130.37001
[10] Umemura, H., Differential Galois theory of infinite dimension, Nagoya Math. J., 144, 59-135 (1996) · Zbl 0878.12002
[11] Umemura, H., Théories asymptotiques et équations de Painlevé, 14, 299-339 (2006) · Zbl 1156.34080
[12] Umemura, H., Differential equations and quantum groups, 9, 269-289 (2007) · Zbl 1356.12006
[13] Umemura, H., Sur l’équivalence des théories de Galois différentielles générales, C. R. Math. Acad. Sci. Paris, 346, 21-22, 1155-1158 (2008) · Zbl 1204.12009 · doi:10.1016/j.crma.2008.09.025
[14] Umemura, H., Differential Equations and Singularities, 60 years of J.M. Aroca, 323 (2010)
[15] Veselov, A. P., Integrable mappings, Russian Math. Surveys, 46, 1-51 (1991) · Zbl 0785.58027 · doi:10.1070/RM1991v046n05ABEH002856
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.