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

TL;DR

This paper investigates the structure of hyperbolic components in polynomial skew-products of ^2, analyzing bifurcation loci and establishing connections with algebraic braids and permutation classes, extending previous quadratic case results to higher degrees.

Contribution

It introduces a homogeneous parametrization for the family, computes the bifurcation accumulation set, and constructs maps linking hyperbolic components to algebraic braids and permutation classes for higher degrees.

Findings

Computed the accumulation set of bifurcation locus boundary.

Constructed a map from hyperbolic components to algebraic braids.

Established a surjective map to conjugacy classes of permutations.

Abstract

We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of $\mathbb{C}^2$ of degree $d\geq2$, 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 $\pi_0(\mathcal{D}')\to AB_d$ 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 $\pi_0(\mathcal{D}')\to\mathrm{Conj}(\mathfrak{S}_d)$ 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 $\pi_0(\mathcal{D}')$.