On the nonexistence of iterative roots

For the germ of a holomorphic mapping F : (U, 0) → (C, 0) of the form F(z) = ρz + ⋯, where ρ is a primitive root of unity of order d⩾2, criteria for the existence of a continuous iterative root of given order and the topological linearizability of F are given. llowing conditions are equivalent: (1) Fd = Id; (2) the germ of the mapping F(z) is topologically conjugate to the germ of the mapping qz; (3) the germ of the mapping F has a continuous iterative root of order dk for every k ⩾ 1. ≠ Id, then for a given positive integer N the germ of the mapping F has a continuous iterative root of order N iff d · gcd(N, ind(Fd, 0) − 1) divides ind(Fd, 0) −1.