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Polynomial semiconjugacies, decompositions of iterations, and invariant curves

We study the functional equation $A\circ X=X\circ B$, where $A,$ $B$, and $X$ are polynomials over $\mathbb C$. Using previous results of the author about polynomials sharing preimages of compact sets, 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 about invariant curves of maps $F:\,\mathbb C^2 \rightarrow \mathbb C^2$ of the form $(x,y)\rightarrow (f(x),f(y))$, where $f$ is a polynomial, and a version of the result of Zieve and Müller about decompositions of iterations of a polynomial.