Real and Complex Dynamical Systems

本文研究了平面多项式自同构的动力学，这些自同构定义了真实的平面和复平面的自同构。 本计画将利用复分析与复几何的方法来研究复映射的动力学。 我们研究的一个重要部分将是了解复杂的结果何时以及在多大程度上可以应用于底层的真实的系统。 我们的主要重点将放在Henon家庭的二次映射的平面。 在这个族中我们研究了马蹄形映射。 除此之外，我们将描述马蹄轨迹的边界形状，以及边界上映射的行为。 我们也将应用我们的方法来描述某些分支的动力学，一个动机是对平面的多项式同态的理解应该是对研究平面的一般多项式同态的动力学有用的指导。 它应该提供映射和分支的例子，可以详细地理解，并提供的例子，作为路标的理论发展更一般的映射。 与此密切相关的是平面的双有理映射的研究。 特别地，我们将分析在晶格动力学中重整化问题中出现的双有理映射。

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.