Normal forms for nonautonomous difference equations

We extend Henry Poincaréʹs normal form theory for autonomous difference equations χk + 1 = f(χk) to nonautonomous difference equations χk + 1 = fk(χk). Poincaréʹs nonresonance condition αj-Пni=1=1αqii≠0 for eigenvalues is generalized to the new nonresonance condition λj αj Пni=1αqii≠0 for spectral intervals.