The topology of low dimensional continua

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。平面连续体X（及其上的映射）通常通过考虑它们在球面上的补来研究。标准的方法是使用从单位圆盘到球面上X的互补分量U的黎曼映射。这种方法有用的原因之一是保角映射编码u边界的几何形状。我们建议使用一种替代这种解析方法的方法，使这种技术更容易应用于拓扑设置和某些极限情况下，解析技术不起作用。可以证明，保形外射线可以用度量定义的外射线代替，从而得到与保形情况相同的素端理论。此外，共形映射可以被度量定义的U划分为不相交的凸集所取代，这也捕获了U边界的几何形状。这种方法已经被用来获得平面不动点问题的一个可能的最小反例的附加信息，并表明平面的所有正取向映射，包括所有全纯映射，在任何不可分离的不变子连续统中都必须有一个不动点。这种方法也很重要，因为它表明，平面连续体的每一个同位素，从同一点开始，都可以扩展到整个平面的同位素。（标准卡拉多核收敛不足以建立这个结果。）我们计划使用这些结果来解决以下问题：平面的每个映射是否在任何非分离不变子连续统中固定一个点，以及每个齐次树状平面连续统是否是伪弧？我们的方法广泛使用了单元磁盘的几何分层概念。复杂动力系统的研究通常是通过分析系统在周期点附近的行为来进行的。对这些点的存在性的研究自然导致不动点的概念。对不动点存在性的研究是一个古老而成熟的数学分支。在这个项目中，我们打算研究平面自身映射下不动点的存在性。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Plane continua X (and maps on them) are often studied by considering their complements in the sphere.The standard approach is to use a Riemann map from the unit disk to a complementary component U of X in the sphere. One of the reasons this approach is useful is that the conformal map encodes the geometry of the boundary of U. We propose to use an alternative to this analytic approach which makes it easier to apply such techniques in the topological setting and certain limit cases where the analytic techniques do not work. It can be shown that conformal external rays can be replaced by metrically defined external rays which leads to the same prime end theory as in the conformal case. Moreover, the conformal map can be replaced by a metrically defined partition of U into disjoint convex sets, which also captures the geometry of the boundary of U. This approach has already been used to obtain additional information about a possible minimal counter example to the plane fixed point problem and to show that all positively oriented maps of the plane, which include all holomorphic maps, must have a fixed point in any non-separating invariant sub-continuum. This approach was also critical in showing that every isotopy of a plane continuum,starting at the identity, can be extended to an isotopy of the entire plane. (Standard Caratheodory kernel convergence is insufficient to establish this result.) We plan to use these results to attack the following problems: does every map of the plane fix a point in any non-separating invariant sub-continuum and is every homogeneous tree-like plane continuum a pseudo arc? Our approach makes extensive use of the notion of a geometric lamination of the unit disk.Complicated dynamical systems are often studied by analyzing the behavior of a system near periodic points. The study of the existence of such points naturally leads to the notion of a fixed point. The study of the existence of fixed points is an old and well established branch of mathematics. In this project we propose to study the existence of fixed points under maps of the plane onto itself.