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
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