The set K^- for hyperbolic non-invertible maps
The set K^- for hyperbolic non-invertible maps
复制标题
双曲不可逆映射的集合 K^-
DOI：
10.1017/s0143385702000445
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发表时间：
2002
影响因子：
0.9
通讯作者：
Eugen Mihailescu
中科院分区：
文献类型：
--
作者：
Eugen Mihailescu
任意公理 \mathbb{P}^2\mathbb{C} 上的全纯映射具有自然遍历测度 \mu。集合 K^- 是具有任意近邻且史前不收敛于 supp \mu 的点的集合。该集合类似于来自 Hénon 映射情况的具有有界向后迭代的点集合。对于 Hénon 微分同胚，Bedford 和 Smillie 证明 K^- 要么有一个空的内部，要么 \mathop{K}\limits^\circ{}^{-}= 有限多个排斥周期点的盆地的并集。对于 \mathbb{P}^2 的 s-双曲全纯自同态，我们在这里证明唯一的可能性是 \mathop{K}\limits^\circ{}^{-} = \emptyset。这回答了福纳斯的问题。我们还证明当自同态是 Hénon 映射的扰动时， \mathop{K}\limits^\circ{}^{-} 是有限多个排斥盆的并集。还讨论了 s 双曲线图的几个重要示例，其中一些来自产品图的扰动，其他来自螺线管。
Any Axiom A holomorphic map on \mathbb{P}^2\mathbb{C} has a natural ergodic measure \mu. The set K^- is the set of points which have arbitrarily close neighbors with prehistories not convergent to supp \mu. This set is the analogue of the set of points with bounded backwards iterates from the case of Hénon mappings. For Hénon diffeomorphisms it was shown by Bedford and Smillie that K^- either has an empty interior or \mathop{K}\limits^\circ{}^{-}= union of the basins of finitely many repelling periodic points. For s-hyperbolic holomorphic endomorphisms of \mathbb{P}^2, we show here that the only possibility is \mathop{K}\limits^\circ{}^{-} = \emptyset. This answers a question of Fornaess. We also prove that \mathop{K}\limits^\circ{}^{-} is the union of finitely many repelling basins when the endomorphism is a perturbation of a Hénon mapping. Several non-trivial examples of s-hyperbolic maps are discussed as well, some of them coming from perturbations of product maps and others from solenoids.