A study of the dynamics of a family of antipolynomials

我们称反多项式族f_c (z) =z^^<-d>+c的连通性轨迹为多重角。我们证明了Julia集在奇周期双曲分量的闭包中连续依赖于Hausdorff度规，因此吸引环的直接盆地收敛于抛物环的直接盆地。我们还证明了抛物线弧线本身不相交，不同抛物线弧线的闭包只在尖点相交，弧线上的0- ecalle高度点是角0及其反角的内射线的落点。利用这些事实，我们证明了临界值映射是开放单元磁盘上d+1度的分支覆盖。我们证明了多角在主双曲分量附近不是局部连通的，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。我们证明了在奇周期双曲分量的边界上，抛物型循环的全纯指标是实数，当参数接近一个尖点时，全纯指标发散到+*，当指标大于1时，双曲分量外出现反全纯分岔。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，并证明了它们的连续性。

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.