Dynamics of postcritically bounded polynomial semigroup II
Dynamics of postcritically bounded polynomial semigroup II
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后临界有界多项式半群 II 的动力学
DOI：
10.1112/jlms/jdt017
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
H. Sumi
中科院分区：
文献类型：
--
作者：
I. Kim;C. Lecuire and K. Ohshika;Ken'ichi Ohshika;H. Sumi
研究了黎曼球面上多项式映射生成的半群的动力学，使得复平面上的后临界集是有界的。此外，我们还研究了多项式的相关随机动力学。此外，我们还研究了具有有界平面后临界集的多项式半群的斜积的纤维动力学。利用均匀纤维拟共形手术的纤维丛，我们表明，如果这样的半群的Julia集是不连通的，那么存在的家庭不可数许多相互不相交的拟圈的一致膨胀参数的康托集，密集的Julia集的半群。此外，给出了纤维Julia集J γ满足J γ是Jordan曲线而不是拟圆，J γ的无界分支是John域，Jγ的有界分支不是John域的充分条件.证明了在一定条件下，随机Julia集几乎必然是Jordan曲线，而不是拟圆。发现并系统地研究了多项式半群和多项式随机动力学中许多在通常的多项式动力学中不存在的新现象。
We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with a bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation that are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia setJγto satisfy thatJγis a Jordan curve but not a quasicircle, the unbounded component ofis a John domain and the bounded component of ℂ \Jγis not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated.