Higher bifurcations for polynomial skew-products
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
Title :
Higher bifurcations for polynomial skew-products
Author(s) :
Astorg, Matthieu [Auteur]
Institut Denis Poisson [IDP]
Bianchi, Fabrizio [Auteur]
Centre National de la Recherche Scientifique [CNRS]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Institut Denis Poisson [IDP]
Bianchi, Fabrizio [Auteur]
Centre National de la Recherche Scientifique [CNRS]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
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]
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 >
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 :
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