Algebraic curves A^{∘ l}(x)-U(y)=0 and arithmetic of orbits of rational functions

2018/01/06 by Fedor Pakovich, Pakovich, Fedor · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1801.01985

openalex publication_date 2018/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a description of pairs of complex rational functions A and U of degree at least two such that for every d≥ 1 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 \mathbb P¹(K) under iterates of A with the value set U(\mathbb P¹(K)), where A and U are rational functions defined over a number field K.

Citations

• Finiteness theorems for commuting and semiconjugate rational functions

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• On the multiplier spectrum of polynomials

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