  Problems of stabilization of the unstable cycle of one-dimensional complex
dynamical system are briefly discussed. These questions reduced to the problem
of description of the ranges of polynomials $q(z) = q_1z + q_2z^2 +\dots +
q_nz^n$ defined in the unit disk and normalized by the conditions $q(1) = 1 $
and this is the main subject of the present paper.
