  Our first main result is a construction of a simple formal normal form for
holomorphic diffeomorphisms in C^n whose differentials have one-dimensional
family of resonances in the first m eigenvalues, m <= n (but more resonances
are allowed for other eigenvalues). Next, we provide invariants and give
conditions for the existence of basins of attraction. Finally, we give
applications and examples demonstrating the sharpness of our conditions.
