Holomorphic Anosov systems
Holomorphic Anosov systems
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全纯阿诺索夫系统
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发表时间：
1995
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通讯作者：
É. Ghys
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作者：
É. Ghys
紧致复流形的全纯复同态群是一个有限维李群，人们可以期望对这种复同态的动力学行为有一个相当完整的描述，至少在低维情况下是这样。例如，紧致K/ihler流形的全纯同态，保持K~ihler形式的上同调类，具有非常简单的动力学，拓扑熵为零[Fu]，[Sn]。F. Enriques描述了一个复曲面，它的全纯复同态群有无穷多个连通分量[En]，[Ro]。然而，确实存在非常有趣的例子，值得注意。一些K3-曲面有一个无限离散的群，其动力学研究似乎很有前途[Maz]。更经典的例子是GL（n，~）的矩阵A保持某个格ACCn，因此导出复环面~n/A的全纯双同态4。如果A的谱与单位圆不相交，则这个自同构/1是Anosov型的，具有丰富的动力学。本文的目的是研究全纯Anosov同构和流的结构，特别是在低维情况下。
The group of holomorphic diffeomorphisms of a compact complex manifold is a finite dimensional Lie group and one could expect a rather complete description of the dynamical behaviour of such diffeomorphisms, at least in low dimensions. For instance, holomorphic diffeomorphisms of compact K/ihler manifolds, preserving the cohomology class of the K~ihler form, have very simple dynamics, with zero topological entropy [Fu], [Sn]. A theorem of F. Enriques describes alyebraic surfaces for which the group of holomorphic diffeomorphisms has infinitely many connected components [En], [Ro]. However, very interesting examples do exist and deserve attention. Some K3-surfaces have an infinite discrete group of diffeomorphisms whose dynamical study seems promising [Maz]. More classical examples are provided by matrices A of GL(n, ~ ) preserving some lattice A C Cn and, therefore, inducing a holomorphic diffeomorphism,4 of the complex torus ~n/A. If the spectrum of A is disjoint from the unit circle, this diffeomorphism/1 is of Anosov type, with rich dynamics. The purpose of this paper is to investigate the structure of holomorphic Anosov diffeomorphisms and flows, especially in low dimensions.