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.

我们称之为家庭f_c（z）= z ^^ <-d>+c的连接基因座，多方人表明，朱莉娅集合在整个奇数周期的超曲式成分封闭的封闭中，相对于hausdorff指标，奇怪的时期的整个奇数群，因此吸引了cycles concles conc cyc cyc cyc cycc cyc cyc cyc cyc cyc cyc cycc cyc cycc cycc cyc cycc cycc cyc cycc cycc cyc cyc cyccecscement的闭合。我们还表明，抛物线弧不会与自身相交，即仅在牙孔点上闭合不同的抛物线弧，即弧上的0-ecalle高度点是角度0的内部射线的陆点及其相反。使用这些事实，我们已经表明，临界值图是开放单元磁盘上D+1的分支覆盖物。我们表明，多层并未在主要的双曲线成分附近局部连接，并且它在最大值的奇数超级抛物性时期的主抛物性弧附近并未在原理抛物线附近连接到对称符号的最大调谐量。 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.