{"title":"Higher bifurcations for polynomial skew products","authors":"M. Astorg, Fabrizio Bianchi","doi":"10.3934/jmd.2022003","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>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 <inline-formula><tex-math id=\"M1\">\\begin{document}$ k $\\end{document}</tex-math></inline-formula> critical points bifurcate <i>independently</i>, with <inline-formula><tex-math id=\"M2\">\\begin{document}$ k $\\end{document}</tex-math></inline-formula> 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.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Modern Dynamics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/jmd.2022003","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
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 \begin{document}$ k $\end{document} critical points bifurcate independently, with \begin{document}$ k $\end{document} 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.
期刊介绍:
The Journal of Modern Dynamics (JMD) is dedicated to publishing research articles in active and promising areas in the theory of dynamical systems with particular emphasis on the mutual interaction between dynamics and other major areas of mathematical research, including:
Number theory
Symplectic geometry
Differential geometry
Rigidity
Quantum chaos
Teichmüller theory
Geometric group theory
Harmonic analysis on manifolds.
The journal is published by the American Institute of Mathematical Sciences (AIMS) with the support of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University.