论文标题
多项式偏斜产物的较高分叉
Higher bifurcations for polynomial skew-products
论文作者
论文摘要
我们继续研究多项式偏斜产品家族的参数空间。 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.对于一维情况，这是一个明显的区别。 证明基于倾斜引理的变体，该变体应用于Misiurewicz参数的后临界集。通过分析标准，用于非分叉电流的自身交流，我们推断出分叉电流支撑的平等和对此类家庭的分叉措施的平等。结合Dujardin和Taflin的结果，这也意味着这些家族中分叉措施的支持具有非空的内部。作为证据的一部分，我们在这些家族中构建了Codimension 1的亚家族1，其中分叉基因座的内部为非内部。这提供了一个新的独立证明，证明了任意大维的霍明型家族的存在，其分叉基因座的内部没有空的内部。最后，它表明，分叉措施支持的豪斯多夫维度在其支持的任何点上都是最大的。
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.
