Multivariable Complex Dynamics

多变量复杂动力学

• 批准号: 9801074
• 负责人: Jeffrey Diller
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 1998-09-17
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801074&HistoricalAwards=false
• 关键词: Multivariable; Complex; Dynamics

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.

迪勒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的双有理映射上。此类映射的混沌集往往会在某些正闭合电流的支持下出现，因此多能理论是理解双有理映射动态的自然工具。 迪勒计划将这一见解更进一步，通过理解一个双有理映射完全根据它对适当选择的正电流的作用。 例如，他建议用一个动态定义的非常数解析圆盘空间的商来代替射影平面，作为双有理映射的自然域。 我们希望，由于细化域将不会有点，这是不确定的地图，这将是一个更自然的设置，用于构建和理解动态显着电流。 动态系统是任何随时间演变的过程，例如天气或互联网上的信息流量。 理解这样的系统的行为通常既非常有用又非常困难。 关于这类系统的问题和解决方案通常是数学的，可以简化为方程组，然后可以通过手工或计算机模拟进行理论分析。 这项工作将有助于进一步了解更复杂和更现实的系统。