Hyperbolic components and iterated monodromy of polynomial skew-products of $\mathbb{C}^2$
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| Format: | Preprint |
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2025
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| author | Tapiero, Virgile |
| author_facet | Tapiero, Virgile |
| contents | We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of $\mathbb{C}^2$ of degree $d\geq2$, with a fixed base $p\in\mathbb{C}[z]$. Using a homogeneous parametrization of the family, we compute the accumulation set $E$ of the bifurcation locus on the boundary of the parameter space. Then in the case $p(z)=z^d$, we construct a map $π_0(\mathcal{D}')\to AB_d$ from the set of unbounded hyperbolic components that do not fully accumulate on $E$, to the set of algebraic braids of degree $d$. This map induces a second surjective map $π_0(\mathcal{D}')\to\mathrm{Conj}(\mathfrak{S}_d)$ towards the set of conjugacy classes of permutations on $d$ letters. This article is a continuation in higher degrees of the work of Astorg-Bianchi in the quadratic case $d=2$, for which they provided a complete classification of the hyperbolic components belonging to $π_0(\mathcal{D}')$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_07484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolic components and iterated monodromy of polynomial skew-products of $\mathbb{C}^2$ Tapiero, Virgile Dynamical Systems Complex Variables 2020: 37F10, 32U40, 37F46, 20E08 We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of $\mathbb{C}^2$ of degree $d\geq2$, with a fixed base $p\in\mathbb{C}[z]$. Using a homogeneous parametrization of the family, we compute the accumulation set $E$ of the bifurcation locus on the boundary of the parameter space. Then in the case $p(z)=z^d$, we construct a map $π_0(\mathcal{D}')\to AB_d$ from the set of unbounded hyperbolic components that do not fully accumulate on $E$, to the set of algebraic braids of degree $d$. This map induces a second surjective map $π_0(\mathcal{D}')\to\mathrm{Conj}(\mathfrak{S}_d)$ towards the set of conjugacy classes of permutations on $d$ letters. This article is a continuation in higher degrees of the work of Astorg-Bianchi in the quadratic case $d=2$, for which they provided a complete classification of the hyperbolic components belonging to $π_0(\mathcal{D}')$. |
| title | Hyperbolic components and iterated monodromy of polynomial skew-products of $\mathbb{C}^2$ |
| topic | Dynamical Systems Complex Variables 2020: 37F10, 32U40, 37F46, 20E08 |
| url | https://arxiv.org/abs/2501.07484 |