The topology of low dimensional continua

这个奖项是根据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)通常是通过考虑它们在球面上的互补来研究的。标准方法是使用从单位圆盘到球面中X的互补分量U的黎曼映射。这种方法有用的原因之一是，保角映射编码了U的边界的几何形状。我们建议使用一种替代这种解析方法的方法，这使得这种方法更容易应用于拓扑环境和某些分析方法不起作用的极限情况。结果表明，共形外光线可以用度规定义的外光线代替，从而得到与共形情形相同的素端理论。此外，共形映射可以用度量定义的U到不相交凸集的划分来代替，这也捕捉了U的边界的几何。这种方法已经被用来获得关于平面不动点问题的一个可能的极小反例的附加信息，并证明了平面上的所有正向映射，包括所有全纯映射，在任何不分离不变子连续体中一定有一个不动点。这一方法在证明平面连续体的每一个从恒等式开始的等价性都可以扩展到整个平面的等价性方面也是至关重要的。(标准Caratheodory核收敛不足以建立这一结果。)我们计划利用这些结果来解决以下问题：平面上的每个映射都在任何非分离不变子连续体中固定一个点吗？每个齐次树形平面连续体都是伪弧吗？我们的方法广泛地利用了单位圆盘的几何叠层的概念，通常通过分析系统在周期点附近的行为来研究复杂的动力系统。对这些点的存在的研究自然导致了固定点的概念。研究不动点的存在是数学中一个古老而成熟的分支。在这个项目中，我们打算研究平面到自身的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.