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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.

我们把反多项式族f_c（z）=z^^ +c的连通性轨迹称为<-d>多角，证明了在奇周期双曲分支的闭域中，Julia集连续依赖于Hausdorff度量，从而吸引圈的直接盆收敛于抛物圈的直接盆.我们还表明，抛物线弧不相交本身，不同的抛物线弧的封闭只相交于尖点，弧上的0-Ecalle高度点是一个土地点的内部射线的角度为0和它的匡威。利用这些事实，我们证明了临界值映射是开单位圆盘上d+1次的分支覆盖.我们证明了多重角在主双曲分量附近不是局部连通的，并且在奇周期最大调谐双曲分量的主抛物弧附近不是局部路径连通的.我们还证明了，在奇周期的双曲分量的边界上，抛物圈的全纯指标是真实的，当参数接近尖点时，全纯指标发散到+*，当全纯指标大于1时，在双曲分支外出现反全纯分支。多项式P_c（z）=z^d+c的周期点的连续性。