Dynamics in Two Complex Variables

PI: John Smillie，康奈尔大学dms -0302357摘要描述系统如何随时间变化的数学模型的研究继续提供基本的数学挑战。当模型是非线性时，这些模型的长期行为或动力学可能是混沌的，并且动力学对模型参数的依赖可能非常微妙。理解解释非线性动力学的基本机制的一个经典方法是在所有映射或微分同态的集合中寻找一般行为；另一种方法是仔细观察精心挑选的特殊家庭。可变动力学的最新进展表明，这些方法可以互补。特别是对复二次族的研究，得到了关于实映射的解析族和光滑单峰族的新结果。PI将研究一类特殊的二维动力系统：两个复维的多项式微分同态。这些问题将从多个角度进行研究。PI将使用来自表面光滑微分同态研究的思想，来自势理论的方法，来自一个复杂变量的动力学方法和计算机工具。希望为从一维复杂动力学到高维真实复杂动力学的思想迁移提供一条途径。对描述系统如何随时间演变的数学模型的理解继续为许多科学领域提供重要的见解。这些模型被用来描述心脏的节律、激光的脉冲和疾病的传播。尽管取得了进展，但在研究这些数学模型方面仍然存在一些基本问题。即使看似简单的模型也会带来巨大的数学困难。在过去的20年里，在使用分形对象（如Julia集合和Mandelbrot集合）来理解具有一个自由度的系统的动力学方面取得了重要进展。本研究的目的是将这些新方法和新概念中的一些引入到理解具有两个自由度的系统的问题上。如果这些想法被证明是富有成效的，那么就会对动力系统领域产生积极的影响。这可能反过来对一系列科学领域产生影响，在这些领域中，时间演化的数学模型发挥着作用。

PI: John Smillie, Cornell UniversityDMS-0302357AbstractThe study of mathematical models that describe how systems change with time continues to provide fundamental mathematical challenges. When the models are nonlinear the long-term behavior, or dynamics, of these models can be chaotic and the dependence of the dynamics on the parameters of the model can be extraordinarily delicate. One classic approach to understanding the fundamental mechanisms that explain nonlinear dynamics is to look for generic behavior in the collection of all maps or diffeomorphisms; another approach is to look carefully at well chosen special families. Recent advances in one variable dynamics show that these approaches can be complementary. In particular the study of the complex quadratic family has led to new results about analytic and smooth unimodal families of real maps. The PI will study a particular family of two-dimensional dynamical systems: polynomial diffeomorphisms in two complex dimensions. These will be studied from many points of view. The PI will use ideas from the study of smooth diffeomorphisms of surfaces, methods from potential theory, methods from dynamics in one complex variable and computer tools. The hope is to provide a pathway for the migration of ideas from one-dimensional complex dynamics to higher dimensional real and complex dynamics.The understanding of mathematical models that describe how systems evolve with time continues to provide essential insights in many areas of science. Such models are used to describe the rhythms of the heart, the pulsing of lasers and the spread of disease. Despite the progress that has been made there are still fundamental problems remaining in the study of such mathematical models. Even seemingly simple models can present formidable mathematical difficulties. In the past 20 years important progress has been made in using ideas connected with fractal objects such as Julia sets and the Mandelbrot set in understanding the dynamics of systems with one degree of freedom. The aim of the proposed research is to bring some of these new methods and concepts to bear on the problem of understanding systems with two degrees of freedom. If these ideas prove fruitful there could be a positive effect on the field of dynamical systems. This could in turn have an influence on a range of scientific fields in which mathematical models of time evolution play a role.