Polynomial semiconjugacies, decompositions of iterations, and invariant curves

We study the functional equation A o X = X o B, where A, B, and X are polynomials with complex coefficients. Using results of [13] about polynomials sharing preimages of compact sets in C, we show that for given B its solutions may be described in terms of the filled-in Julia set of B. On this base, we prove a number of results describing a general structure of solutions. The results obtained imply in particular the result of Medvedev and Scanlon [10] about invariant curves of maps F: ℂ² → ℂ² of the form (x, y) → (f(x), f(y)), where f is a polynomial, and a version of the result of Zieve and Müller [22] about decompositions of iterations of a polynomial.