Special regular polynomial skew products

TL;DR

This paper introduces a class of special polynomial skew products in two complex variables, characterizes them through conjugation and algebraic group relations, and generalizes known one-variable polynomial results.

Contribution

It defines and characterizes 'special' regular polynomial skew products, linking their structure to algebraic groups and number fields, extending classical one-variable polynomial theory.

Findings

Special skew products are conjugate to maps involving power, Chebyshev, or Dickson polynomials.

They are semiconjugate to affine maps on algebraic groups.

All multipliers of these maps lie in a fixed number field.

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.