Holomorphic motions of hyperbolic sets
Holomorphic motions of hyperbolic sets
复制标题
双曲集的全纯运动
DOI：
10.1307/mmj/1030132192
复制
发表时间：
1998
影响因子：
0.9
通讯作者：
Mattias Jonsson
中科院分区：
文献类型：
--
作者：
Mattias Jonsson
设M是复Hermitian流形，{fa}a∈D是M的自同态全纯族，其中D是单位圆盘.这意味着由（a，x）→ fa（x）定义的映射D×M → M是全纯的。设f = f0有一个紧满射不变子集K，即f（K）= K。例如，K可以是不动点或周期轨道，但也可以是更复杂的集合，如有理函数的Julia集。然后我们可以问K在映射f的扰动fa下是否持久。例如，如果K是的一个不动点，那么我们问a是否有一个不动点Ka在K附近。一个充分的（虽然不是必要的）条件是不动点K是双曲的，这意味着f在K处的导数没有模为1的特征值。对于一般的集合K，有一个自然的双曲性概念。让我们首先考虑映射fa是单同态的情况。精确的定义（例如可以在[R]中找到）在这里就不讲了，但是它说K上的切丛连续分裂成两个不变子丛，在这两个不变子丛上f的导数分别膨胀和收缩。真实的动力学的一个基本结果是双曲集在映射f中的扰动下是持久的（见[R]）。在我们的例子中，这意味着如果a s足够小，则fa有一个接近K的双曲集Ka，并且存在一个接近单位共轭f的同胚ha| K到FA| Ka .如果K是一个双曲不动点，那么根据隐函数定理，fa的不动点Ka全纯依赖于a。这一概念自然地推广到更一般的集合K上，就是全纯运动的概念，它的定义在第1节中给出。
Let M be a complex Hermitian manifold and {fa}a∈D a holomorphic family of endomorphisms of M,whereD is the unit disk. This means that the map D×M → M, defined by(a, x)→ fa(x), is holomorphic. Suppose that f = f0 has a compact surjectively invariant subset K, that is,f(K) = K. For example, K could be a fixed point or a periodic orbit, but also a more complicated set such as the Julia set of a rational function. We may then ask if K is persistent under the perturbationfa of the mapf. For instance, ifK is a fixed point of , then we ask if a has a fixed pointKa nearK for a small enough. A sufficient (albeit not necessary) condition for this is that the fixed point K be hyperbolic, meaning that the derivative of f atK has no eigenvalue of modulus 1. There is a natural notion of hyperbolicity for general sets K. Let us first consider the case when the maps fa are diffeomorphisms. The precise definition (which can be found e.g. in [R]) will not be stated here, but it says that the tangent bundle overK splits continuously into two invariant subbundles on which the derivative of f is expanding and contracting, respectively. One basic result in real dynamics is that hyperbolic sets are persistent under perturbations in the map f (see [R]). In our case this means that if a s small enough, thenfa has a hyperbolic set Ka close toK, and there exists a homeomorphism ha close to the identity conjugating f |K to fa|Ka . If K is a hyperbolic fixed point, then it follows from the implicit function theorem that the fixed point Ka of fa depends holomorphically on a. The natural generalization of this to more general sets K is the notion of a holomorphic motion, the definition of which is given in Section 1.