## 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.

### MSC:

 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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### References:

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