Inverse images of polynomial mappings and polynomials orthogonal on them

Let T be a polynomial with complex coefficients. First, we study the inverse images of the real and imaginary axes under a polynomial mapping T in detail. Then for an arbitrary polynomial ρ and a sequence (pn) of orthogonal polynomials the orthogonality behaviour of the sequence of polynomials (ρ(pn∘T))n∈N is investigated. In particular necessary and sufficient conditions are given such that (ρ(pn∘T))n∈N is a subsequence of polynomials orthogonal with respect to a positive measure supported on a compact subset of the real line.