Infinitely many periodic attractors for holomorphic maps of $2$ variables
Infinitely many periodic attractors for holomorphic maps of $2$ variables
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$2$ 变量的全纯映射的无限多个周期性吸引子
DOI：
10.2307/2951819
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发表时间：
1997
影响因子：
4.9
通讯作者：
G. Buzzard
中科院分区：
文献类型：
--
作者：
G. Buzzard
离散动力系统研究中的一个重要发展是纽豪斯使用持久同宿切线证明了紧致曲面的一个大的C2同态集合有无穷多个共存的周期吸引子或汇[6]，其中“大”指的是一个开集同态的剩余子集。在本文中，我们得到了这一结果的各种空间的全纯映射的两个变量。纽豪斯后来扩展了他的结果，证明了在任何具有同宿切线的曲面单同态附近都存在这样的剩余集[7]。最近，Palis和维亚纳将后者的结果推广到更高的维度，当稳定流形的余维为1时[10]，Romero使用鞍代替汇[12]获得了更高余维稳定流形的类似结果。然而，在每一种情况下，这种构造都简化为对线上相交的康托集的研究：在适当的投影下，基本集的稳定和不稳定流形映射到线上的康托集，这些流形之间的切线对应于康托集的交点。切线的一般展开然后产生周期性吸引子或鞍。这种线性康托集的简化在很大程度上取决于只有一个扩展本征值的事实。甚至罗梅罗的结果更高余维稳定流形涉及减少这种情况。在全纯的情况下，特征值以共轭对的形式出现（从真实的观点来看），所以这种简化是不可能的。相反，稳定流形和不稳定流形是黎曼曲面，并且在将基本集合的稳定流形和不稳定流形扩展为叶理之后，这些叶理将在（真实的）2维圆盘中相切，并且稳定流形和不稳定流形对应于该圆盘中的康托集。因此，基本集之间的持久相切对应于平面上两个康托集的稳定相交。
An important development in the study of discrete dynamical systems was Newhouse's use of persistent homoclinic tangencies to show that a large set of C2 diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors, or sinks [6], where "large" refers to a residual subset of an open set of diffeomorphisms. In the present paper, we obtain this result for various spaces of holomorphic maps of two variables. Newhouse later extended his result to show that such residual sets exist near any surface diffeomorphism which has a homoclinic tangency [7]. More recently, Palis and Viana extended this latter result to higher dimensions when the stable manifold has codimension one [10], and Romero obtained an analogous result for higher codimension stable manifolds using saddles in place of sinks [12]. In each case, however, the construction reduces to the study of intersecting Cantor sets in the line: under an appropriate projection, the stable and unstable manifolds of a basic set are mapped to Cantor sets in the line, and a tangency between these manifolds corresponds to a point of intersection of the Cantor sets. The generic unfolding of tangencies then gives rise to periodic attractors or saddles. This reduction to linear Cantor sets depends heavily on the fact that there is only one expanding eigenvalue. Even Romero's result for higher codimension stable manifolds involves a reduction to this case. In the holomorphic setting, eigenvalues come in conjugate pairs (from a real point of view), so this reduction is not possible. Instead, stable and unstable manifolds are Riemann surfaces, and after extending the stable and unstable manifolds of a basic set to foliations, these foliations will be tangent in a (real) 2-dimensional disk, and the stable and unstable manifolds correspond to Cantor sets in this disk. Hence, persistent tangencies between basic sets correspond to the stable intersection of two Cantor sets in the plane.