On Planar Mappings with Invariant Continua

建议：DMS-0072626摘要。这个建议涉及保持某些次连续不变的平面的映射。在历史上，保定向同胚得到了平面映射的第一个重要结果。平面上的所有同胚要么是保定向的，要么是逆定向的。在这个方案中，我们将引入一类新的定向映射。面向一切的地图要么是正面的，要么是负面的。我们期望证明平面上的开映射和单调映射的所有合成都是定向的。我们推广了关于非分离不变平面连续体中平面到正向映射族的同胚的不动点存在性的已知结果。我们预计这对所有定向地图都是正确的。我们将建立这些问题与平面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的结构。我们将利用原始技术以及连续体理论和动力学系统的结果。

Proposal: DMS-0072626Abstract.This proposal is concerned with mappings of the plane which keepcertain subcontinua invariant. Historically the first significantresults for maps of the plane were obtained for orientationpreserving homeomorphisms. All homeomorphisms of the plane areeither orientation preserving or orientation reversing. In thisproposal we will introduce a new class of oriented mappings. Everyoriented map is either positively or negatively oriented. Weexpect to show that all compositions of open and monotone (properand onto) mappings of the plane are oriented. We have generalizedknown results on the existence of fixed points in non-separatinginvariant plane continua for homeomorphisms of the plane to theclass of positively oriented maps. We expect this to be true forall oriented maps. We will establish connections between theseproblems and the structure of invariant continua of branchedcovering maps of the plane including the structure of the boundaryof invariant simply connected domains. We will make use of primeend techniques as well as results from continuum theory anddynamical systems.