Mathematical Sciences: The Topological Structure of Planar Continua

本研究主要围绕着一类弱可链平面连续体(伪弧的连续映象的平面连续体)展开。假设伪弧是非常非局部连通的，这类连续体包括许多病态连续体以及所有局部连通连续体。作者证明了它包括所有不分离的齐次连续体，Lelek证明了它包括所有可链的连续体，Minc证明了这类中的所有不分离连续体都具有不动点性质。加藤已经证明，没有可链接的连续体允许扩张的同态。由此产生的一个自然问题是，弱可链平面连续体是否允许扩张同胚。我们将建立这个问题与平面上分支覆盖映射的不变连续体的结构之间的联系，包括不变单连通区域的边界结构。我们将利用最好的结束技术以及来自连续统理论和动力系统的结果。该方案涉及齐次平面连续体的分类，非分离平面连续体的不动点性质，以及复多项式的非局部连通Julia集的拓扑结构。平面连续体在许多领域都有应用，最显著的是作为动力系统中的不变集。动力系统是一个研究随时间变化的系统的数学模型的性质的数学领域。不变连续体通常决定动力系统的长期行为，尤其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，最早可以追溯到1800年末的S。尽管在过去的80年里取得了重大进展，但仍有一些长期存在的问题尚未解决。在这项建议中，我们将在其中一些旧问题和从其他数学领域产生的新问题之间建立联系。我们将使用来自邻近数学领域的旧思想和新工具和技术的混合来指出这些问题的可能解决方案。我们将把重点放在这些问题的拓扑方面，尽管解决办法很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.