A study of the dynamics of a family of antipolynomials

反多项式族动力学研究

基本信息

批准号：
07640258
负责人：
NAKANE Shizuo
金额：
$ 1.47万
依托单位：
Tokyo Institute of Polytechnics
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1997
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07640258/
关键词：
antipolynomial multicorn hyperbolic component Julia set Ecalle height bifurcation holomorphic index Grotzsch defect hyperbolic components parabolic arcs critical value map 内射線 muHicorn external ray 非弧状連結性反正則分岐

项目摘要

We call the connectedness locus of the family f_c (z) =z^^<-d>+c of antipolynomials the multicorn.We have shown that Julia sets depend continuously with respect to Hausdorff metric throughout the closure of hyperbolic components of odd periods, hence that immediate basins of attracting cycles converge to those of parabolic cycles. We have also shown that parabolic arcs do not intersect themselves, that closures of distinct parabolic arcs intersect only at cusp points, that the 0-Ecalle height point on the arc is a land point of an internal ray of angle 0 and its converse. Using these facts, we have shown that critical value maps are branched coverings of degree d+1 over the open unit disk.We have shown that the multicorn is not locally connected near the main hyperbolic component and that it is not locally pathwise connected near the principal parabolic arcs of maximally tuned hyperbolic components of odd periods not on the arcs of symmetry.We have shown that, on the boundary of hyperbolic components of odd periods, the holomorphic indices of parabolic cycles are real and diverge to +* as the parameter approaches a cusp point and antiholomorphic bifurcation occurs outside hyperbolic components if the index is greater than 1.We have calculated the Grotzsch defects of fixed points and 2-periodic points of polynomials P_c (z) =z^d+c and have shown their continuity.

我们将反多项式族 f_c (z) =z^^<-d>+c 的连通性轨迹称为多角。我们已经证明，在奇数周期双曲分量的闭包过程中，Julia 集连续依赖于 Hausdorff 度量，因此，吸引周期的直接盆地会收敛于抛物线周期的盆地。我们还证明了抛物线弧本身并不相交，不同抛物线弧的闭合仅在尖点处相交，弧上的 0-Ecalle 高度点是角度为 0 及其逆角的内射线的着陆点。利用这些事实，我们证明了临界值图是开单位圆盘上 d+1 度的分支覆盖。我们证明了多角形在主双曲分量附近不是局部连通的，并且在主双曲分量附近也不是局部路径连通的。奇数周期最大调谐双曲分量的主抛物线不在对称弧上。我们已经证明，在奇数周期双曲分量的边界上，抛物线周期的全纯指数是实数并且发散到 +* 作为如果指数大于 1，则参数接近尖点，反全纯分岔出现在双曲分量之外。我们计算了多项式 P_c (z) =z^d+c 的不动点和 2-周期点的 Grotzsch 缺陷，并显示他们的连续性。