Dynamics of Polynomial Diffeomorphisms

最近动力系统中一些最令人印象深刻的工作是对一个复杂变量的动力学研究的产物。提出这一建议的基本假设是，当维数大于1时，复杂方法在动力学中也起着重要作用。这一建议的主题是动力系统的一个特殊的“模型族”，多项式微分同态在两个复杂的维度。复Henon微分同胚是一个值得注意的特例。这可能是最简单的可逆全纯动力系统，具有有趣的动力学。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，到Misurewicz映射，半双曲和Collet-Eckmann映射。膨胀性质的双变量类比是双曲性。这一建议的重点是理解半双曲条件的两个变量类比，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有关。在两个变量中，临界点的概念需要用稳定流形和不稳定流形的正则性等概念来代替。拟双曲微分同态的一个有趣的例子是极大熵的实多项式微分同态，如实Henon情况下马蹄铁的极限。虽然在以前的工作中已经回答了关于这种微分同构行为的几个问题，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，它可以用来描述单个昆虫种群在连续年份中的行为。当数学模型是线性的，就有一个完善的基础理论。当模型是非线性的，有一些重要的理论问题我们还不能解决。近年来，数学家在理解逻辑图方面取得了重大突破。一个被证明必不可少的技术是考虑相关的复杂动力系统。我的建议解决了当涉及的系统维度大于1时使用类似复杂技术的一些问题（例如，当存在两个相互作用的人群时）。

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.)