Multivariable Complex Dynamics

Diller提出了复射影空间上有理映射的迭代研究，主要集中在射影平面的双象映射上。这种地图的混沌集往往出现在某些正闭合电流的支持下，因此多能理论是理解两国地图动力学的自然工具。迪勒计划将这一见解进一步深入，从正确选择的正电流的角度完全理解两国地图。例如，他建议用一个动态定义的非恒定解析盘空间的商来代替投影平面，作为双区映射的自然域。我们的希望是，由于精炼的域不会有地图上不确定的点，它将是一个更自然的环境，用于构建和理解动态重要的电流。动态系统是任何随时间演化的过程——例如，天气或互联网上的信息流量。理解这类系统的行为通常既非常有用，又非常困难。有关这类系统的问题和解决方案通常是数学的，可以简化为一组方程，然后可以用手工进行理论分析或用计算机模拟进行实验分析。这项工作将有助于进一步理解更加复杂和更加现实的系统。

Diller proposes to study the iteration of rational maps on complex projective space, focusing primarily on birational maps of the projective plane. Chaotic sets for such maps tend to occur as the support of certain positive closed currents, thus pluripotential theory is a natural tool to use in understanding the dynamics of birational maps. Diller plans to take this insight a step further, by understanding a birational map entirely in terms of its action on appropriately chosen positive currents. For instance, he proposes to replace the projective plane by a dynamically defined quotient of the space of non-constant analytic disks as the natural domain of a birational map. The hope is that since the refined domain will not have points which are indeterminate for the map, it will be a more natural setting for constructing and understanding the dynamically significant currents. A dynamical system is any process that evolves in time-for instance, the weather or information traffic on the internet. It is often both very useful and very difficult to understand the behavior of such systems. The problems and solutions concerning such systems are often mathematical and can be reduced to sets of equations which can then be analyzed theoretically by hand or experimentally by computer simulation. This work will contribute to the further understanding of still more complicated and ever more realistic systems.