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.
Gotou, Dynamical systems of correspondences on the projective line II: degrees of multiplier maps
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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.