The topology of low dimensional continua

低维连续体的拓扑

• 批准号: 0906316
• 负责人: Lex Oversteegen
• 金额: $ 13.32万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906316&HistoricalAwards=false
• 关键词: topology; low; dimensional; continua

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.

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。平面连续统X（以及其上的映射）的研究通常考虑它们在球面中的补，标准的方法是使用从单位圆盘到球面中X的一个补分量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的边界几何。 我们建议使用这种分析方法的替代方案，这使得它更容易应用这种技术在拓扑设置和某些限制的情况下，分析技术不工作。它可以表明，共形外部光线可以取代度量定义的外部光线，导致相同的素端理论在共形的情况下。此外，保角映射可以被替换为度量定义的划分， U的边界的几何形状，这也捕获了U的边界的几何形状。这种方法已经被用来获得额外的信息，一个可能的最小反例的平面不动点问题，并表明，所有积极定向的地图的平面，其中包括所有全纯映射，必须有一个不动点在任何非分离不变子连续统。这种方法也是至关重要的，它表明平面连续统的每一个合痕，从单位元开始，都可以扩展到整个平面的合痕。（标准Caratheodory核收敛不足以建立这个结果。我们计划使用这些结果来攻击以下问题：是否每个地图的平面固定在任何非分离不变的子连续统的一个点，是每一个均匀的树状平面连续的伪弧？我们的方法广泛地利用了单位圆盘的几何层积的概念，复杂的动力系统通常通过分析系统在周期点附近的行为来研究。研究这些点的存在性自然会引出不动点的概念。 研究不动点的存在性是一个古老而完善的数学分支。在这个项目中，我们建议研究在平面到自身的映射下不动点的存在性。