Fatou components with punctured limit sets
Fatou components with punctured limit sets
复制标题
带刺破极限设置的 Fatou 组件
DOI：
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发表时间：
2013
影响因子：
0.9
通讯作者：
Han Peters
中科院分区：
文献类型：
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作者：
Luka Boc;J. Fornæss;Han Peters
研究了$mathbb{P}^{2}$中全纯自同态的不变Fatou分支。在递归情况下，这些分量由Fornæss和Sibony[某些全纯映射的递归域的分类]来分类。数学课。安。301(4)(1995)，813-820]。Ueda[射影空间上的全纯映射和Fatou映射的延拓.密歇根数学J.56(1)(2008)，145-153]通过证明极限集不可能是被穿孔的圆盘来完成这一分类。最近，Lyubich和Peters[耗散Hénon映射的不变Fatou分量的分类]。Preprint]在极限集唯一的附加假设下，对非循环不变Fatou分量进行了分类。再一次，这种分类的所有可能性都是已知的，除了被刺穿的椎间盘。这里我们证明了被穿透的圆盘确实可以作为非循环Fatou分量的极限集出现。我们给出了具有非循环Fatou分支的$mathbb{C}^{2}$的全纯自同态和多项式自同态的许多例子，其轨道收敛到任意解析集的正则部分。
We study invariant Fatou components for holomorphic endomorphisms in $mathbb{P}^{2}$. In the recurrent case these components were classified by Fornæss and Sibony [Classification of recurrent domains for some holomorphic maps. Math. Ann. 301(4) (1995), 813–820]. Ueda [Holomorphic maps on projective spaces and continuations of Fatou maps. Michigan Math J.56(1) (2008), 145–153] completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and Peters [Classification of invariant Fatou components for dissipative Hénon maps. Preprint] classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of $mathbb{C}^{2}$ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.