Hyperbolicity and bifurcations in holomorphic families of polynomial skew products

TL;DR

This paper studies the stability and bifurcations of holomorphic families of polynomial skew products in complex dynamics, classifies hyperbolic components, and introduces a new equidistribution result for bifurcation currents in higher dimensions.

Contribution

It provides the first equidistribution theorem for bifurcation currents in higher-dimensional holomorphic dynamical systems and classifies hyperbolic components in polynomial skew product families.

Findings

Hyperbolicity is preserved under dynamical stability in these families.

Complete classification of hyperbolic components analogous to the Mandelbrot set.

Describes the geometry of bifurcation locus and current near parameter space boundary.

Abstract

We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation current. This is established in the general setting of families of endomorphisms of $\mathbb{P}^k$ and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one.