For general rational functions A of degree m >= 2, decompositions of iterates A^n are unique up to equivalence, implying product maps on CP1 x CP1 have non-trivial periodic curves iff the component functions are conjugate.
Pakovich, The algebraic curve P (x) − Q(y) = 0 and functional equations, Complex Var
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Periodic curves for general endomorphisms of $\mathbb C\mathbb P^1\times \mathbb C\mathbb P^1$
For general rational functions A of degree m >= 2, decompositions of iterates A^n are unique up to equivalence, implying product maps on CP1 x CP1 have non-trivial periodic curves iff the component functions are conjugate.