Higher bifurcations for polynomial skew-products

Matthieu Astorg (1) , Fabrizio Bianchi (2)

1 IDP - Institut Denis Poisson
2 LPP - Laboratoire Paul Painlevé - UMR 8524

We continue our investigation of the parameter space of families of polynomial skew-products. We study the self-intersections of the bifurcation current, and in particular the bifurcation measure, through the simultaneous bifurcations of multiple critical points. Our main result is the equality of the supports of the bifurcation current and the bifurcation measure for families of polynomial skew-products over a fixed base. This is a striking difference with respect to the one-dimensional case. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior. It also provides a new 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. Our proof is based on an analytical criterion for the non-vanishing of the bifurcation currents and on a geometric method to create multiple bifurcations at a common parameter. The latter is a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter.

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Matthieu Astorg, Fabrizio Bianchi. Higher bifurcations for polynomial skew-products. 2020. ⟨hal-02882110v1⟩