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

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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 of \(({{\mathbb {P}}}^1({{\mathbb {C}}}))^2\) of the form \((z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).\) In particular, we show that if \(A\in {{\mathbb {C}}}(z)\) is not a “generalized Lattès map”, then any (A, A)-invariant curve has genus zero and can be parametrized by rational functions commuting with A. As an application, for A defined over a subfield K of \( {{\mathbb {C}}}\) we give a criterion for a point of \(({{\mathbb {P}}}^1(K))^2\) to have a Zariski dense (A, A)-orbit in terms of canonical heights, and deduce from this criterion a version of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits. We also prove a result about functional decompositions of iterates of rational functions, which implies in particular that there exist at most finitely many \((A_1, A_2)\)-invariant curves of any given bi-degree \((d_1,d_2).\)

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Acknowledgements

The author would like to thank Dragos Ghioca, Laura DeMarco, Thomas Tucker, and Junyi Xie for helpful conversations.

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  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, 8410501, Israel

    Fedor Pakovich

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  1. Fedor Pakovich
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Correspondence to Fedor Pakovich.

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Communicated by Wei Zhang.

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This research was supported by ISF Grant No. 1432/18.

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Pakovich, F. Invariant curves for endomorphisms of \({{\mathbb {P}}}^1\times {{\mathbb {P}}}^1\). Math. Ann. 385, 259–307 (2023). https://doi.org/10.1007/s00208-021-02304-5

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