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.
The Moduli Space of Cubic Rational Maps
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abstract
We construct the moduli space, $M_d$, of degree $d$ rational maps on $\mathbb{P}^1$ in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for $M_3$. Using classical invariant theory, we give solutions to the following problems: (1) explicitly construct, from a moduli point $P\in M_d(k)$, a rational map $\phi$ with the given moduli; (2) find a model for $\phi$ over the field of definition (i.e. explicit descent). We work out the method in detail for the cases $d=2,3$.
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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.