Special regular polynomial skew products

Abstract

We define a regular polynomial skew product $(p(z),q(z,w))$ of $\mathbb{C}^2$ of degree $d\geq 2$ to be special if it is triangularly conjugate to a map of the form $(p(z),q(w))$, where $p$ and $q$ are power maps or $\pm$Chebyshev maps, or of the form $(z^d,D_d(w,\zeta z^m))$, where $\zeta^{d-1}=1$, $m\in\{1,2\}$, and $D_d$ is the Dickson polynomial of degree $d$. We justify this definition by showing the following equivalence. (1) $f$ is special. (2) $f$ is semiconjugate to an affine self-map $g$ in skew product form of a 2-dimensional connected and commutative algebraic group $G$ over $\mathbb{C}$. (3) All multipliers of $f$ are contained in a fixed number field $K$. This generalizes the one-variable polynomial case.