On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials
On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials
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DOI：
10.1017/etds.2015.65
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发表时间：
2014-04
影响因子：
0.9
通讯作者：
S. Mukherjee;S. Nakane;D. Schleicher
中科院分区：
文献类型：
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作者：
S. Mukherjee;S. Nakane;D. Schleicher
多角是单临界反全纯多项式$\bar{z}^{d}+c$的连通轨迹。我们调查的双曲成分的边界的结构：我们证明，从双曲成分的偶数周期的分支结构是一个期望的地图，全纯依赖于一个复杂的参数（例如，作为Mandelbrot集，在这种设置，这是一个不明显的事实），而在双曲成分的奇数周期的分支结构是非常不同的。特别地，奇周期双曲分量的边界仅由抛物参数组成，并且双曲分量之间沿沿着整个弧存在分叉，但分叉比仅为2。我们还计算了多角形的任意周期的双曲分量的个数。由于反全纯多项式仅实解析地依赖于参数，本文中使用的大多数技术与用于证明全纯设置中的相应结果的技术有很大不同。
The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^{d}+c$ . We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.