Invariant curves for endomorphisms of $${{\mathbb {P}}}^1\times {{\mathbb {P}}}^1$$

Let $$A_1, A_2\in {{\mathbb {C}}}(z)$$ be rational functions of degree at least two that are neither Lattès maps nor conjugate to $$z^{\pm n}$$ or $$\pm T_n.$$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms $$({{\mathbb {P}}}^1({{\mathbb {C}}}))^2$$ the form $$(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$$ In particular, we show if $$A\in is not a “generalized map”, then any (A, A)-invariant curve has genus zero can parametrized by commuting with A. As an application, A defined over subfield K $$ {C}}}$$ give criterion point {P}}}^1(K))^2$$ have Zariski dense A)-orbit in terms canonical heights, deduce from this version conjecture Zhang on existence points forward orbits. also prove result about functional decompositions iterates functions, which implies particular there exist most finitely many $$(A_1, A_2)$$ -invariant given bi-degree $$(d_1,d_2).$$