Analytic Coordinates Recording Cubic Dynamics
Analytic Coordinates Recording Cubic Dynamics
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解析坐标记录立方动力学
DOI：
10.1201/b10617-14
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
L. Tan
中科院分区：
文献类型：
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作者：
C. Petersen;L. Tan
设H是三次多项式的中心双曲分量（即包含z7 → z3的分量）。Milnor的工作表明H具有一种普适性。我们构造了一个解析坐标H记录的动力学不变量。我们使用二次动力学作为参数模型。该坐标至少对H的大部分边界具有良好的扩展性质。我们通过证明其中的一些扩展属性来说明这一点。本文将有理映射的参数空间视为迭代动力系统。有理映射族的参数空间通常可以自然分解为双曲轨迹和非双曲轨迹：有理映射是双曲的当且仅当它的所有临界点都被吸引到吸引周期轨道上。双曲轨迹是开的（并且在任何合理的族中是稠密的）。双曲分量是双曲轨迹的连通分量。在同一个双曲分量内的映射具有本质上相同的宏观动力学。然而，当我们移动到双曲分量的边界时，动力学的这种定量变化就变成了定性的，而且往往是剧烈的。因此，该领域的一个主要研究兴趣是了解双曲分量的边界结构，包括其拓扑结构，几何结构和发生的动力学分叉。给定这样一个分量H，为了研究它的边界，第一步实际上是详细研究H的内部结构，以便有效地测量H内部动力学的（无穷小）变化。解决这个问题的一种方法是找到一个合适的模型空间X和一个映射Φ：H → X，同时满足以下性质：1.（坐标）Φ在H中是单射的，并且Φ（f）解析地依赖于f ; 2.（dynamics）Φ（f）记录了f的动力学不变量的完整集合。换句话说，从这些不变量（连同一些组合数据）可以重建f的共形共轭。一旦我们有了这样一个坐标，我们就可以用Φ（H）来表示H，用φ Φ（H）来研究φ H。这将取决于Φ的边界延伸性质。因此，下一步将是研究Φ或Φ−1如何延伸到边界，以及它在多大程度上仍然反映动力学不变量。2000年数学学科分类：37 F10、37 F20、37 F45、32 H 02。
Let H be the central hyperbolic component of cubic polynomials (i.e. the one containing z 7→ z3). Works of Milnor show that H enjoys a kind of universality property. We construct an analytic coordinate on H recording the dynamical invariants. We use quadratic dynamics as parameter models. This coordinate has good extension properties to at least a large part of the boundary of H. We illustrate this by proving some of these extension properties. This paper concerns parameter spaces of rational maps viewed as dynamical systems through iteration. The parameter space of a family of rational maps has often a natural decomposition into the hyperbolic locus and the non-hyperbolic locus: a rational map is hyperbolic if and only if all its critical points are attracted to attracting periodic orbits. The hyperbolic locus is open (and conjecturally dense, in any reasonable family). A hyperbolic component is a connected component of the hyperbolic locus. Maps within the same hyperbolic component have essentially the same macroscopic dynamics. However, this quantitative change of dynamics becomes qualitative and often drastic when one moves to the boundary of the hyperbolic component. Therefore a major research interest in this field is to understand the boundary structure of a hyperbolic component, including its topology, geometry and the bifurcations of the dynamics that occurs. Given such a component H , in order to study its boundary, the first step is actually to study in detail the inner structure of H , to put our hand on an effective measuring of the (infinitessimal) changes of the dynamics within H . One way to address this problem is to find some suitable model space X together with a map Φ : H → X , satisfying simultaneously the following properties: 1. (coordinate) Φ is injective in H and Φ(f) depends analytically on f ; 2. (dynamics) Φ(f) records a complete set of dynamical invariants of f . In other words from these invariants (together with some combinatorial data) one can reconstruct f up to conformal conjugacy. Once we have such a coordinate, we may use Φ(H) to represent H and use ∂Φ(H) to study ∂H . This will depend on boundary extension properties of Φ. Thus the next step will be to study how well Φ or Φ−1 extends to the boundary and to which extent it still reflects the dynamical invariants. ∗AMS 2000 Mathematics Subject Classifications: 37F10, 37F20, 37F45, 32H02.