Ensembles d'unicite pour les polynomes (Sets of Uniqueness for   polynomials)
  Let E be a compact set of C of positive logarithmic capacity. Let us suppose that for every polynomial $P\not=id$ we have $P^{-1}(E)\not=E$. Then for all no constant polynomials f and g such that $f^{-1}(E)=g^{-1}(E)$ we have f=g. 
Tien-Cuong Dinh