Complex Dynamics in Higher Dimensions

摘要作者提出研究紧复流形亚纯自映射的动力学问题。如果这样的映射扩展了流形的某个同调类，那么主要的思想是应该有一个不同的不变电流表示这个类。不变电流之间的交叉点应反过来产生不变测度，以描述轨道的分布，其中轨道的作用是最复杂的。因此，自纯映射对上同调的线性作用对于确定该映射的点向动力学的非概率性图象有很大的帮助。为了证明这些想法，研究者建议使用复杂几何、光滑遍历理论和多能分析的技术组合。物理和数学中的各种问题都归结为关于特定亚纯映射的动力学问题。研究者建议使用其中的一些作为他研究的起点和测试案例。在过去的几十年里，动力系统的研究变得越来越重要。简而言之，动力系统是任何东西——例如：天气、股票市场、太阳系等等——这些都是根据（至少在理论上）明确的、可编纂的规则在时间上进化的。研究动力系统的一般目标是在给定其当前状态的情况下对其未来状态作出预测。如果未来非常遥远，那么正如任何诚实的预测者都会承认的那样，这通常是相当困难的。系统当前状态的微小变化可能意味着最终结果的巨大差异。这里提出的研究致力于更好地理解动力系统中的不稳定性——何时发生，为什么发生，以及如何识别它。该研究还致力于通过对不稳定系统的最终行为进行统计预测，而不依赖于对其当前状态的完全理解，从而实现对不稳定性的部分控制。研究者提议研究的动力系统本质上是数学性质的，但它们也是人们分析与现实世界现象有关的系统的原型。其思想是，天气、股市和太阳系不应被视为独立的、完全不相关的动力系统，而应被视为相同普遍现象的特殊例子，并应进行类似的分析。**********************************

DMS-0140408Complex Dynamics in Higher DimensionProposal AbstractThe principal investigator proposes to study the dynamics of meromorphicself-maps of compact complex manifolds. If such a map expands somecohomology class of the manifold, then the main idea is that there oughtto be a distinguished invariant current representing this class.Intersections between invariant currents should in turn give rise toinvariant measures that describe the distribution of orbits where theaction of the map is most complicated. Hence the linear action of ameromorphic map on cohomology should go a long way toward determining aprobabilistic picture of the pointwise dynamics of the map. To justify these ideas, the investigator proposes to use a combination of techniquesfrom complex geometry, smooth ergodic theory, and pluripotentialanalysis. Various problems in physics and mathematics reduce to questionsabout the dynamics of particular meromorphic maps. The investigatorproposes to use some of these as starting points and test cases for hisresearch.The study of dynamical systems has grown increasingly important in thepast several decades. Put briefly, a dynamical system is anything--e.g. the weather, the stock-market, the solar system, etc--that evolves intime according (at least theoretically) to definite and codifiable rules.The general goal in studying a dynamical system is to make predictionsabout its future state given its present one. If the future is veryfar off then this, as any honest forecaster will admit, is typically quite difficult. Small changes in the present state of the system can imply huge differences in eventual outcome. The research proposed here isdevoted to better understanding instability in dynamical systems--whenit occurs, why it occurs, and how to recognize it. The research is alsodevoted to attaining partial control over instability by makingstatistical predictions about the eventual behavior of an unstable systemthat do not depend on perfect understanding of its present state. Thedynamical systems that the investigator proposes to study are mathematicalin nature, but they are also archtypes for the systems that one analyzesin connection with real-world phenomena. The idea is that weather andthe stockmarket and the solar system ought not be treated as a separateand totally unrelated dynamical systems but as particular examplesof the same general phenomena and subject to similar kinds of analyses.**********************************