Multivariable Complex Dynamics

Diller建议研究复射影空间上有理映射的迭代，主要集中在射影平面上的双调映射。这类映射的混沌集往往是作为某些正闭合电流的支撑而出现的，因此多势理论是用来理解双态映射动力学的自然工具。迪勒计划将这一洞察力更进一步，完全根据其对适当选择的正向电流的作用来理解一张出生地图。例如，他建议用动态定义的非常数解析圆盘空间的商来代替射影平面，作为二元映射的自然域。人们希望，由于精化后的区域不会有地图上不确定的点，它将是构建和理解动态显著电流的更自然的环境。动力系统是任何随时间演变的过程--例如，互联网上的天气或信息流量。理解这类系统的行为通常既非常有用，又非常困难。与这种系统有关的问题和解决方案通常是数学的，可以归结为一组方程，然后可以手工进行理论分析，或者通过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复杂、更现实的系统。

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.