Mathematical Sciences: Embeddings of Continua

数学科学：Continua 的嵌入

• 批准号: 9400945
• 负责人: Ira Wayne Lewis
• 金额: $ 6万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-01-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400945&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Embeddings; Continua

9400945 Lewis Lewis will investigate a number of problems related to embeddings of continua in manifolds and other spaces with a uniform structure. Techniques involving inverse limits, accessibility, and prime end structure will be studied. He is seeking characterizations of when various inverse limits can be embedded in the plane and when homeomorphisms of continua can be extended to homeomorphisms of the plane. He is also investigating continua with the property that all of their nondegenerate proper subcontinua are equivalently embedded, or which admit rigid embeddings in the plane, and various homogeneous embeddings in real 3-space. Determining the existence of an indecomposable continuum which is almost homogeneously embedded in the plane will be important in the study of dynamics. Questions relating embeddings and extensions of maps to computation of Hausdorff dimension and to attractors and cofrontiers are among dynamic topics related to this project. Lewis hopes to narrow the currently large gap between known theorems and existing examples in the theory of continua by broadening the base of the theory and looking for generic classes of examples and construction techniques. Recent work in discrete dynamics indicates that considerations of embeddings and extensions of maps are crucial in identifying which sets can be attractors or closures of orbits, how sensitive dynamic properties are to perturbation, and what type dynamics is possible or even generic. These investigations should yield more information about the continua themselves and new ways of approaching problems which until now have proven slow to yield to attacks. Possibilities for new approaches lie in the areas of decompositions, homogeneity, characterizations of spaces, and investigations of subcontinua. ***

小刘易斯9400945 刘易斯将调查一些问题有关的嵌入连续流形和其他空间的统一结构。 将研究涉及逆极限，可访问性和总理结束结构的技术。 他正在寻求刻画时，各种逆限制可以嵌入在平面上，当同胚的连续可以扩展到同胚的平面。 他还调查连续的财产，所有的非退化适当subcontinua等效嵌入，或承认刚性嵌入在平面上，和各种均匀嵌入在真实的3空间。 在动力学研究中，确定一个几乎均匀嵌入平面的不可分解连续统的存在性是很重要的。 有关嵌入和扩展的问题，以计算的Hausdorff维数和吸引子和cofrontiers的地图是在动态的主题与这个项目有关。 刘易斯希望通过拓宽理论的基础，寻找实例和构造技术的通用类，来缩小连续统理论中已知定理和现有实例之间目前存在的巨大差距。 最近的工作在离散动力学表明，考虑嵌入和扩展的地图是至关重要的，在确定哪些集可以吸引或关闭的轨道，如何敏感的动态特性是扰动，什么类型的动力学是可能的，甚至通用。 这些调查应该产生更多关于连续体本身的信息，以及解决问题的新方法，这些问题到目前为止已经证明是缓慢的。 新方法的可能性在于分解、齐性、空间的特征化和子连续统的研究。 ***