## Abstract

Let A_{1}, A_{2}∈ C(z) be rational functions of degree at least two that are neither Lattès maps nor conjugate to z^{±}^{n} or ± T_{n}. We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of (P1(C))2 of the form (z_{1}, z_{2}) → (A_{1}(z_{1}) , A_{2}(z_{2})). In particular, we show that if A∈ 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 C we give a criterion for a point of (P1(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}).

Original language | English |
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Pages (from-to) | 259-307 |

Number of pages | 49 |

Journal | Mathematische Annalen |

Volume | 385 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Feb 2023 |

## ASJC Scopus subject areas

- General Mathematics

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