On intertwined polynomials

Abstract

Let $A_1$ and $A_2$ be polynomials of degree at least two over $\mathbb{C}$. We say that $A_1$ and $A_2$ are intertwined if the endomorphism $(A_1,A_2)$ of $\mathbb{C}{\mathbb{P}}^1\times \mathbb{C}{\mathbb{P}}^1$ given by $(z_1,z_2)\mapsto (A_1(z_1),A_2(z_2))$ admits an irreducible periodic curve that is neither a vertical nor a horizontal line. We denote by $\mathrm{Inter}(A)$ 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 $\mathrm{Inter}(A)$. We also obtain a bound on the possible periods of periodic curves for endomorphisms $(A_1,A_2)$ in terms of the sizes of the symmetry groups of the Julia sets of $A_1$ and $A_2$.