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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.01877 |
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Table of Contents:
- 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 $A_2$.