Separatrices for non solvable dynamics on \(\mathbb{C},0\) (Q1337038)

We define the separatrices for pseudogroups of diffeomorphisms of open neighbourhoods of the origin in the complex plane \(\mathbb{C}\) and prove their existence for non solvable pseudogroups (Theorem 1). This extends a result by \textit{A. A. Shcherbakov} [Mosc. Univ. Math. Bull. 37, No. 4, 10- 16 (1982); translation from Vestn. Mosk. Univ., Ser. I 1982, No. 4, 10-15 (1982; Zbl 0517.30009)] accurately. Our method also applies to prove the topological rigidity theorem for generic pseudogroups attributed to Shcherbakov [Tr. Semin. Im. I. G. Petrovskogo 10, 170-196 (1984; Zbl 0568.30010)].