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.


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


Zbl 1033.32020
Full Text: DOI EuDML