Voronoi-cell criterion for selecting a root permutation

Let the roots of the corresponding univariate polynomials be all distinct, and let $\mathbf{a}$ be a vector whose entries are points in the complex plane. Let $F_{\mathbf{x},\mathbf{a}}$ be the induced vector mapping defined by the composition iteration, and let the root-induced Voronoi partition of the complex plane be formed using some metric. Voronoi-cell conjecture. The iteration expresses the particular permutation of the roots on the complex plane if each entry of $\mathbf{a}$ lies in a distinct Voronoi cell of that partition. This would explain how the choice of $\mathbf{a}$ breaks the symmetry among the $n!$ possible solutions when the roots are distinct.