## Galoisian envelope of a rational map of $$\mathbb P^1$$. (Enveloppe Galoisienne d’une application rationnelle de $$\mathbb P^1$$.)(French. English summary)Zbl 1137.37022

The Galois envelope of a dynamical system has been defined by B. Malgrange [Monogr. Enseign. Math. 38, 465–501 (2001; Zbl 1033.32020)]. In particular, in case $$R:\mathbb P^1\to \mathbb P^1$$ is a rational map, its Galois envelope is defined as the minimal Lie groupoid of ideal sheaves of differential equations on $$\mathbb P^1$$ for which $$R$$ is a solution. Such an envelope is nontrivial in case it is not reduced to $$(0)$$.
The aim of the paper under review is to characterize (up to pre-composition with homographes) all rational maps having a nontrivial Galois envelope. In particular the author shows that all such maps are monomials, or Chebyshev polynomials or the so-called Lattès examples.

### MSC:

 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 53C10 $$G$$-structures 58H05 Pseudogroups and differentiable groupoids

### Keywords:

holomorphic dynamics; Lie groupoid

Zbl 1033.32020
Full Text:
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