Mathematical Sciences: The Topological Structure of Planar Continua

本课题围绕弱链平面连续体（即伪弧的连续像）这类问题展开研究。鉴于伪弧是非局部连通的，这类包括许多病理连续体和所有局部连通的连续体。提出者证明了它包含了所有不可分离齐次连续体，Lelek证明了它包含了所有可链连续体，Minc证明了该类中所有不可分离连续体都具有不动点性质。加藤证明了链连续体不允许扩张同胚。一个自然产生的问题是，弱链平面连续体是否承认扩张同胚。我们将建立这个问题与平面分支覆盖映射的不变连续体结构之间的联系，包括不变单连通域的边界结构。我们将利用素端技术以及连续统理论和动力系统的结果。本文研究了齐次平面连续体的分类、非分离平面连续体的不动点性质以及复数多项式的非局部连通Julia集的拓扑结构问题。平面连续体有许多应用，最显著的是作为动力系统中的不变量集。动力系统是研究随时间变化的系统的数学模型的性质的数学领域。不变连续常常决定一个动力系统的长期行为，特别是在这些系统表现出混沌行为的情况下。对这种连续体的研究历史悠久，可以追溯到19世纪晚期。尽管在过去80年中取得了重大进展，但一些长期存在的问题仍未得到解决。在这个建议中，我们将在这些老问题和从其他数学领域产生的新问题之间建立联系。我们将指出这些问题的可能解决方案，使用来自相邻数学领域的旧思想和新工具和技术的混合。我们将重点关注这些问题的拓扑方面，尽管解决方案可能会对邻近领域的工作产生影响。

This research project is centered around the class of weakly chainable plane continua (plane continua which are the continuous image of the pseudo arc). Given that the pseudo arc is very non-locally-connected, this class includes many pathological continua as well as all locally connected continua. The proposer has shown that it includes all non-separating homogeneous continua, Lelek has shown that it includes all chainable continua and Minc has shown that all non separating continua in this class have the fixed point property. Kato has shown that no chainable continuum admits an expansive homeomorphism. A natural question which arises is whether weakly chainable plane continua admit an expansive homeomorphism. We will establish connections between this problem and the structure of invariant continua of branched covering maps of the plane, including the structure of the boundary of invariant simply connected domains. We will make use of prime end techniques as well as results from continuum theory and dynamical systems. The problems in this proposal are related to the classification of homogeneous plane continua, the fixed point property of non-separating plane continua, and the topological structure of non-locally-connected Julia sets of complex polynomials. Planar continua arise in many applications, most notably as invariant sets in dynamical systems. Dynamical systems is an area of mathematics which studies properties of mathematical models of systems which change over time. Invariant continua often determine the long term behavior of a dynamical system and are particularly of interest in case these systems exhibit chaotic behavior. The study of such continua has a long history dating back to the late 1800's. Despite significant advances during the last 80 years, several long-standing problems remain unresolved. In this proposal we will establish connections between some of these old problems and new ones which arise from other areas of mathematics. We will indicate possible solutions to th ese problems using a mixture of old ideas and new tools and techniques from adjacent mathematical fields. We will focus on the topological aspects of these problems although solutions may well have an impact on work in adjacent fields.