Multivariable complex dynamics

这一建议涉及几个复杂变量、复杂代数几何和动力系统之间的交叉问题。这些问题源于一个非常一般的程序，用于构造和分析投影空间的有理自映射的最大熵测度。这项工作将从保留亚纯二形式的有理映射的特殊情况开始并由其指导。要研究的具体问题包括映射度增长的“代数稳定性”，余维大于1的不变电流的层流性，以及如何动态交叉自然闭合电流以产生不变测度。一般来说，动力系统是任何“东西”。天气、细菌群落、经济或太阳系——它们都按照明确的数学规则随时间演化。给定系统的当前状态，人们常常想知道它的未来状态。下个世纪地球会显著变暖吗？太阳系会分崩离析吗？减税或政府干预的长期金融影响是什么？从广义的数学意义上讲，所有这些问题都归结为理解动力系统的某些方面是“稳定”还是“不稳定”。也就是说，它是否随着系统的发展而缓慢而可预测地变化，或者它是否倾向于随着系统的微小变化而迅速而混乱地变化。这里提出的研究旨在更好地理解动力系统的数学，特别是确定和描述系统中最不稳定的部分。该提案的资金还将支持PI目前建议的两名博士生。它的广泛影响将通过PI参与圣母大学高中数学教师的暑期项目和河湾数学中心来感受，这是一个致力于促进各级数学教育的当地独立非营利组织。

This proposal concerns problems in the intersection between several complex variables, complex algebraic geometry and dynamical systems. The problems stem from a very general program for constructing and analyzing measures of maximal entropy for rational self-maps of projective space. The work will begin with and be guided by the particular case of rational maps preserving a meromorphic two form. Specific issues to be investigated include `algebraic stability' for degree growth of maps, laminarity of invariant currents in codimension greater than one, and how to intersect dynamically natural closed currents to produce invariant measures. A dynamical system, at its most general, is any 'thing'-e.g. the weather, a bacteria colony, an economy, or a solar system-that evolves in time according to definite mathematical rules. Given the present state of the system, one often wants to know something about its future state. Will the earth warm significantly in the next century? Will the solar system fly apart? What are the long term financial effects of a given tax cut or government intervention? In a broad mathematical sense, all of these questions reduce to understanding whether some aspect of a dynamical system is 'stable' or 'unstable'. That is, does it vary slowly and predictably as the system evolves, and or is it prone to change rapidly and chaotically with a small variation in the system. The research proposed here aims at better understanding mathematics of dynamical systems, and in particular at determining and describing those parts of a system which are most unstable. Funding for this proposal will also support the two PhD students the PI is currently advising. It's broader impact will be felt through the PI's involvement with a Notre Dame summer program for high school math teachers and with the Riverbend Math Center, which is a local independent non-profit organization dedicated to promoting math education at all levels.