Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators

We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of "projections" of the iterates on this subspace is determined by the arithmetic properties of the leading eigenvalues.