## Abstract

We give a description of pairs of complex rational functions A and U of degree at least two such that for every d1 the algebraic curve A^{◦d} (x) −U(y) = 0 has a factor of genus zero or one. In particular, we show that if A is not a “generalized Lattès map”, then this condition is satisfied if and only if there exists a rational function V such that U ◦ V = A^{◦l} for some l ≥ 1. We also prove a version of the dynamical Mordell–Lang conjecture, concerning intersections of orbits of points from P^{1} (K) under iterates of A with the value set U(P^{1} (K)), where A and U are rational functions defined over a number field K.

Original language | English |
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Pages (from-to) | 153-183 |

Number of pages | 31 |

Journal | Moscow Mathematical Journal |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2020 |

## Keywords

- Dynamical Mordell
- Lang conjecture
- Riemann surface orbifolds
- Semiconjugate rational functions
- Separated variable curves

## ASJC Scopus subject areas

- Mathematics (all)

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