Geometry and Ergodic Theory of Rational Maps

有理图的几何和遍历理论

• 批准号: 0653678
• 负责人: Jeffrey Diller
• 金额: $ 11.68万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653678&HistoricalAwards=false
• 关键词: Geometry; Ergodic; Theory; Rational; Maps

The principal goal of this proposal is to understand the dynamics of rational self-maps of complex manifolds in terms of the induced linear actions on cohomology groups. The idea is to realize expanding eigenvectors geometrically as invariant positive closed currents carried by the manifold and, by intersecting currents, to produce and understand measures of maximal entropy for the maps. It is also proposed to push beyond ergodic theory for some families of rational maps with extra geometric structure and give more detailed, pointwise descriptions of dynamics.Mathematical laws for an evolving system typically prescribe the way the system changes. If the present state of the system is known, then such laws tell one what happens in the following instant. However, the distant future of the system, while in principle predictable from the same laws, is in practice often impossible to know: computations involved in making predictions can be overwhelming, or the computations can amplify small uncertainties in data about the present, so that predictions become hopelessly vague. Miraculously in such situations, one can often employ less direct means to obtain a rough probabilistic picture of the future. For instance, detailed knowledge of today's weather and the laws of physics will allow one to know the particulars of tomorrow's weather with some confidence, but they will avail one little in predicting the weather a year from tomorrow. Nevertheless, knowing only that today's date is Jan 1st, one can reasonably bet that it will be chill! y a year from today in Minneapolis. Understanding trends of this sort from a mathematical standpoint is called ergodic theory, and that is the subject of the research described in this proposal. It is devoted particularly to ergodic theory of systems known as `rational maps'. Such systems are cleaner and more abstract than weather forecasting, but they are nevertheless very rich and applicable in practical situations where computers are used to solve complicated equations or to optimize the outcome of a particular process.

本文的主要目的是根据上同调群上的诱导线性作用来理解复流形的有理自映射的动力学。我们的想法是将特征向量以几何方式扩展为流形携带的不变正闭合电流，并通过交叉电流来产生和理解映射的最大熵度量。对于一些具有额外几何结构的有理图族，提出了超越遍历理论的建议，并给出了更详细的、逐点的动力学描述。演化系统的数学规律通常规定了系统变化的方式。如果系统的当前状态是已知的，那么这些定律就会告诉我们下一个瞬间会发生什么。然而，系统的遥远未来，虽然原则上可以从同样的定律预测，但在实践中往往是不可能知道的：在进行预测时涉及的计算可能是压倒性的，或者计算可以放大当前数据中的小不确定性，因此预测变得绝望地模糊。不可思议的是，在这种情况下，人们往往可以使用不太直接的方法来获得未来的粗略概率图景。例如，对今天天气和物理定律的详细了解将使人们有信心知道明天天气的细节，但它们在预测一年后的天气方面用处不大。然而，只知道今天是1月1日，人们就可以合理地打赌，今天会很冷！一年后的今天在明尼阿波利斯从数学的角度理解这种趋势被称为遍历理论，这就是本提案中所描述的研究主题。它特别致力于被称为“理性映射”的系统的遍历理论。这样的系统比天气预报更清晰，更抽象，但它们仍然非常丰富，适用于计算机用于解决复杂方程或优化特定过程结果的实际情况。