An iteration problem - Bekbaev, Ural (Institute of Physics Publishing (UK), 2013)
Let F stand for the feld of real or complex numbers, φ : F^n→ F^n be any given polynomial map of the form φ(x) = x + "higher order terms". We attach to it the following operator D : F[x]→ F[x] defined by D(f) = f-f\circleφ, where F[x] = F[x_1; x_2; ...; x_n]- the F-algebra of polynomials in variables x_1; x_2;...; x_n, f \in F[x] and \circle stands for the composition(superposition) operation. It is shown that trajectory of any f\in F[x] tends to zero, with respect to a metric, and stabilization of all trajectories is equivalent to the stabilization of trajectories of x_1; x_2;...; x_n.