Galoisian envelope of a rational map of \(\mathbb P^1\) (Q2458374)

The Galois envelope of a dynamical system has been defined by \textit{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.