On intertwined polynomials
Fedor Pakovich
TL;DR
The paper addresses intertwined polynomials by linking invariant and periodic curves of endomorphisms on $({\mathbb C}{\mathbb P}^1)^2$ to semiconjugacy relations between polynomials. It proves Favre and Gauthier's conjecture, showing that for each degree $d$ there is a finite family of building blocks $B_i$ such that any $B$ intertwined with some iterate of $A$ is conjugate to a twisted iterate of one of the $B_i$, with a uniform bound $r(d)$. A key technical component is a detailed analysis of polynomial semiconjugacies via Engström–Ritt theory, elementary transformations, and Julia set dynamics, including a bound on the periods of irreducible periodic curves given by the lcm of symmetry-related quantities $N(A)=|\Sigma(A)|\varphi(|\Sigma(A)|)$. The results yield a concrete description of the Inter(A) set in terms of iterates of finitely many building blocks and provide period bounds in terms of the symmetry groups of the Julia sets. Together, these findings illuminate how Julia set symmetries govern semiconjugacy structures and periodic dynamics in polynomial maps on projective spaces, with implications for invariant and periodic curves in complex dynamics.
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 $A_2$.