A-stabilization and the ranges of complex polynomials on the unit disk
This is a theoretical contribution to complex analysis and dynamical systems, but the abstract lacks concrete results or comparisons, making it incremental for specialists.
The paper addresses the problem of describing the ranges of complex polynomials on the unit disk under normalization conditions, motivated by stabilization of unstable cycles in one-dimensional complex dynamics. The results provide a characterization of these ranges.
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.