Higher bifurcations for polynomial skew-products

TL;DR

This paper studies polynomial skew-products, showing that near bifurcation points, multiple critical points can bifurcate independently, revealing complex bifurcation structures and maximal Hausdorff dimension of the bifurcation measure support.

Contribution

It demonstrates the existence of high-dimensional parameter regions with independent critical point bifurcations and establishes the equality of bifurcation current and measure supports.

Findings

Multiple critical points bifurcate independently near bifurcation parameters.

Supports of bifurcation current and measure coincide.

Bifurcation locus has non-empty interior in large-dimensional families.

Abstract

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate \emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior.As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.