Special regular polynomial skew products
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910127740682240 |
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| author | Zhang, Yugang |
| author_facet | Zhang, Yugang |
| contents | 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 $(z^d,D_d(w,ζz^m))$, where $ζ^{d-1}=1$, $m\in\{1,2\}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12173 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Special regular polynomial skew products Zhang, Yugang Dynamical Systems Algebraic Geometry Complex Variables 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 $(z^d,D_d(w,ζz^m))$, where $ζ^{d-1}=1$, $m\in\{1,2\}$, 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. |
| title | Special regular polynomial skew products |
| topic | Dynamical Systems Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2604.12173 |