Geometry and Ergodic Theory of Rational Maps

这个建议的主要目标是理解复杂流形的有理自映射的动力学在上同调群上的诱导线性作用。这个想法是实现几何扩展的特征向量作为不变的正封闭电流进行的流形，并通过交叉电流，以产生和理解的措施，最大熵的地图。 它还提出了推动超越遍历理论的一些家庭的合理的地图与额外的几何结构，并给予更详细的，逐点描述的dynamics.Mathematical法律为一个不断发展的系统通常规定的方式系统的变化。 如果系统的当前状态是已知的，那么这样的定律就告诉人们在接下来的瞬间会发生什么。然而，虽然从原则上讲，系统的遥远未来可以从相同的定律中预测，但实际上往往是不可能知道的：进行预测所涉及的计算可能是压倒性的，或者计算可以放大关于现在的数据中的微小不确定性，因此预测变得不可救药地模糊。在这种情况下，人们往往可以奇迹般地采用不那么直接的方法来获得未来的粗略概率图。例如，对今天天气和物理定律的详细了解将使一个人对明天天气的细节有一定的信心，但它们对预测一年后的天气没有什么帮助。 尽管如此，只知道今天的日期是1月1日，人们可以合理地打赌，这将是寒冷的！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，这就是本提案中描述的研究主题。 它特别致力于遍历理论的系统被称为“理性地图”。 这些系统比天气预报更清晰、更抽象，但它们仍然非常丰富，适用于使用计算机求解复杂方程或优化特定过程结果的实际情况。

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.