Dynamics of Polynomial Diffeomorphisms

最近在动力系统中一些最令人印象深刻的工作是对一个复变量的动力学研究的产物。这个提议的基本假设是，当维数大于1时，复杂方法在动力学中也起着重要的作用。 这一建议的主题是一个特殊的“模型家庭”的动力系统，在两个复杂的维度多项式同构。复Henon同态是一个著名的特例。这些可能是最简单的可逆全纯动力系统与有趣的动力学。单变量动力学的一个教训是，有一系列的动力学行为，从扩展映射开始，继续到米苏维茨映射、半双曲映射和科莱-埃克曼映射。 膨胀性质的二元类比是双曲性。这个建议的重点是理解两个变量的模拟半双曲条件，我们称之为准双曲。在一个变量中，这些条件与临界点的递归性质有关。 在二元中，临界点的概念需要用其他概念来代替，如稳定流形和不稳定流形的正则性。一个有趣的例子是最大熵的真实的多项式同构，例如真实的Henon情形中的马蹄铁极限。虽然在以前的工作中已经回答了几个关于这种非同态行为的问题，但仍然存在一些悬而未决的问题。计算机的引入增加了简单的确定性数学模型在许多科学中的有用性。一个说明性的例子是逻辑图，它可以用来描述一个单一的昆虫种群在连续几年的行为。当数学模型是线性的时，有一个发展良好的基础理论。当模型是非线性的，有重要的理论问题，我们还没有能够解决。近年来，数学家在理解逻辑斯蒂映射方面取得了重大突破。一种被证明是必不可少的技术是考虑一个相关的复杂动力系统。我的建议解决了当所涉及的系统的维数大于1时（例如，当有两个相互作用的种群时）使用类似的复杂技术的一些问题。

Some of the most impressive recent work in dynamical systems has been an outgrowth of the study of dynamics in one complex variable. The fundamental assumption motivating this proposal is that complex methods have an important role to play in dynamics when the number of dimensions is greater than one as well. The subject of this proposal is a particular "model family" of dynamical systems, the polynomial diffeomorphisms in two complex dimensions. The complex Henon diffeomorphisms which are a notable special case. These are perhaps the simplest invertible holomorphic dynamical system with interesting dynamics. One of the lessons of dynamics in one variable is that there is a range of dynamical behaviors starting with expanding maps and continuing with Misurewicz maps, semi-hyperbolic and Collet-Eckmann maps. The two variable analog of the expanding property is hyperbolicity. This proposal is focused on understanding the two variable analog of the semi-hyperbolic condition which we call quasi-hyperbolicity. In one variable these conditions are related to the recurrence properties of critical points. In two variables the notion of critical point needs to be replaced by other concepts such as regularity of stable and unstable manifolds. An interesting example of quasi-hyperbolic diffeomorphisms are real polynomial diffeomorphisms of maximal entropy such as limits of horseshoes in the real Henon case. Though several questions about the behavior of such diffeomorphisms have been answered in previous work a number of open questions remain.The introduction of the computer has increased the usefulness of simple deterministic mathematical models in a number of sciences. An illustrative example is the logistic map which can be used to describe the behavior of a single insect population in successive years. When the mathematical model is linear there is a well developed underlying theory. When the model is non-linear there are important theoretical questions which we have not yet been able to address. Recently mathematicians have made important breakthroughs in understanding the logistic map. One technique which proved essential was the consideration of an associated complex dynamical system. My proposal addresses some of the problems of using similar complex techniques when the system involved has dimension greater than one (for example when there are two interacting populations.)