Mathematical Sciences: The Topological Structure of Planar Continua

本研究课题主要围绕弱可链平面连续统（伪弧的连续像平面连续统）展开。假设伪弧是非常非局部连通的，这个类包括许多病态连续统以及所有局部连通连续统。提出者证明了它包括所有非分离的齐次连续统，Lelek证明了它包括所有可链连续统，Minc证明了该类中所有非分离连续统具有不动点性质。加藤证明了没有可链连续统容许扩张同胚。由此产生的一个自然问题是弱可链平面连续统是否允许扩张同胚。我们将建立这个问题与平面的分支覆盖映射的不变连续统的结构之间的联系，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。该方案中的问题涉及齐次平面连续统的分类、非分离平面连续统的不动点性质以及复多项式的非局部连通Julia集的拓扑结构。 平面连续统出现在许多应用中，最显著的是作为动力系统中的不变集。动态系统是一个数学领域，它研究系统的数学模型随时间变化的特性。不变连续统通常决定动力系统的长期行为，并且在这些系统表现出混沌行为的情况下特别感兴趣。对这种连续统的研究有着悠久的历史，可以追溯到19世纪末。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然没有得到解决。在这个建议中，我们将在这些老问题和其他数学领域出现的新问题之间建立联系。我们将指出可能的解决方案，这些问题的混合使用旧的想法和新的工具和技术，从相邻的数学领域。我们将专注于这些问题的拓扑方面，虽然解决方案可能会对相邻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.