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

TL;DR

This work extends the stability and combinatorial classification of hyperbolic components from quadratic polynomial skew-products to all degrees $d\ge2$ in the family $Sk(p,d)$. By employing a homogeneous parametrization, the authors identify the accumulation set $E$ of the bifurcation locus at infinity and show that for $p(z)=z^d$ unbounded hyperbolic components not accumulating on $E$ correspond to algebraic braids of degree $d$, yielding a well-defined map to the braid set $AB_d$ and further to conjugacy classes of permutations in $\mathfrak{S}_d$. Central to the construction is the iterated monodromy group, which is computed for admissible parameters and used to demonstrate braid invariance across hyperbolic components; this provides a two-dimensional analogue of the shift locus and a bridge between holomorphic dynamics in $\mathbb{C}^2$ and braid theory. The authors also present a detailed topological description of the small Julia set as a suspension and establish isotopy results for components of fixed points, enabling a robust classification via $AB_d$ and a surjective map to $\mathrm{Conj}(\mathfrak{S}_d)$. Overall, the paper generalizes the quadratic Astorg-Bianchi classification to higher degrees and deepens the connection between bifurcation theory, iterated monodromy, and braid theory in holomorphic dynamics.

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 $π_0(\mathcal{D}')\to AB_d$ 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 $π_0(\mathcal{D}')$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Example 1.5
• Example 1.6
• Example 1.7
• Theorem 2.1: Astorg-Bianchi AstBian23
• Lemma 2.2
• Definition 2.3