Hyperbolic components and iterated monodromy of polynomial skew-products of ${\mathbb{C}}^2$

Abstract

We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of ${\mathbb{C}}^2$ of degree $d\ge 2$, 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 ${\pi }_0({\mathcal{D}}^{\prime })\to A{B}_d$ 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 ${\pi }_0({\mathcal{D}}^{\prime })\to \mathrm{Conj}({\mathfrak{S}}_d)$ 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 ${\pi }_0({\mathcal{D}}^{\prime })$.

38 pages, 5 figures