On intertwined polynomials

2025/10/02 by Fedor Pakovich, Pakovich, Fedor
Mathematics · #Advanced Mathematical Identities #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2510.01877

openalex publication_date 2025/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A₁ and A₂ be polynomials of degree at least two over \mathbb C. We say that A₁ and A₂ are intertwined if the endomorphism (A₁, A₂) of \mathbb C\mathbb P¹ × \mathbb C\mathbb P¹ given by (z₁, z₂) ↦ (A₁(z₁), A₂(z₂)) admits an irreducible periodic curve that is neither a vertical nor a horizontal line. We denote by Inter(A) the set of all polynomials B such that some iterate of B is intertwined with some iterate of A. In this paper, we prove a conjecture of Favre and Gauthier describing the structure of Inter(A). We also obtain a bound on the possible periods of periodic curves for endomorphisms (A₁, A₂) in terms of the sizes of the symmetry groups of the Julia sets of A₁ and A₂.

Citations

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• Arithmetic properties of families of plane polynomial automorphisms