Title :

Higher bifurcations for polynomial skew-products

Author(s) :

Astorg, Matthieu [Auteur]
Institut Denis Poisson [IDP]
Bianchi, Fabrizio [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Centre National de la Recherche Scientifique [CNRS]

Journal title :

Journal of modern dynamics

Pages :

69

Publisher :

American Institute of Mathematical Sciences

Publication date :

2022

ISSN :

1930-5311

HAL domain(s) :

Mathématiques [math]/Systèmes dynamiques [math.DS]
Mathématiques [math]/Variables complexes [math.CV]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

ULNE
Centre Européen pour les Mathématiques, la Physique et leurs Interactions
Nouveaux points de vue en dynamique rationnelle à plusieurs variables

Comment :

26 pages.

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL