Real and Complex Dynamical Systems

研究平面多项式微分同态的动力学性质。这些映射定义了实平面和复平面的微分同态。这个项目将使用复杂分析和复杂几何的方法来研究复杂映射的动力学。我们研究的一个重要部分将是了解何时以及在多大程度上复杂的结果可以应用于潜在的真实系统。我们主要关注的是平面的Henon族二次映射。在这个家族中，我们研究马蹄形映射。除其他事项外，我们将描述马蹄形轨迹边界的形状，以及边界上映射的行为。我们还将应用我们的方法来描述某些分岔的动力学。一个动机是，对平面的多项式微分同态的理解应该是研究平面的一般微分同态动力学的有用指南。它应该提供可以详细理解的映射和分岔的例子，并提供作为更一般映射理论发展的路标的例子。与此密切相关的是对平面的两国映射的研究。特别地，我们将分析在点阵统计力学的重整化问题中出现的双国家映射。

We investigate the dynamics of plane polynomial diffeomorphisms.These mappings define diffeomorphisms of the real plane as well asthe complex plane. This project will investigate the dynamics ofthe complex mappings using the methods of complex analysis andcomplex geometry. An important part of our research will be tounderstand when and to what extent the complex results can beapplied to the underlying real system. Our main focus will be onthe Henon family of quadratic mappings of the plane. And withinthis family we study the horseshoe mappings. Among other things, wewill describe the shape of the boundary of the horseshoe locus, and the behavior of the mappings on the boundary. Wewill also apply our methods to describe the dynamics of certain bifurcations.One motivation is that an understanding of the polynomialdiffeomorphisms of the plane should be a useful guide to theinvestigation of the dynamics of general diffeomorphisms of theplane. It should provide examples of mappings and bifurcations thatcan be understood in detail and provide the examples thatserve as guideposts for the development of the theory for moregeneral mappings. Closely related to this is the study of birationalmappings of the plane. In particular, we will analyze birationalmaps that arise in renormalization questions in lattice statisticalmechanics.