Geometry and Ergodic Theory of Rational Maps

这一建议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。其思想是将展开的特征向量几何地实现为流形携带的不变正闭流，并通过与流相交来产生和理解映射的最大熵度量。还建议超越某些具有额外几何结构的有理映射族的遍历理论，给出更详细的、逐点的动力学描述。演化系统的数学定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，那么这样的定律告诉人们在接下来的瞬间会发生什么。然而，该系统遥远的未来，尽管原则上可以从相同的定律中预测，但实际上往往是不可能知道的：做出预测所涉及的计算可能是压倒性的，或者计算可能放大关于当前数据的微小不确定性，从而使预测变得无可救药地模糊。奇迹般地，在这种情况下，人们经常可以使用不那么直接的方法来获得对未来的大致概率图景。例如，对今天的天气和物理定律的详细了解将使一个人有一定的信心了解明天的天气细节，但它们在预测明天一年后的天气方面将略有帮助。尽管如此，只要知道今天是1月1日，人们就可以合理地打赌天气会变冷！一年后的今天在明尼阿波利斯。从数学的角度理解这类趋势被称为遍历理论，这也是本提案中所描述的研究主题。它特别致力于被称为“有理映射”的系统的遍历理论。这种系统比天气预报更干净、更抽象，但它们仍然非常丰富，适用于使用计算机求解复杂方程或优化特定过程的结果的实际情况。

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.