On Planar Mappings with Invariant Continua

建议:dms - 0072626 -抽象。本文讨论了平面上保持一定次连续不变的映射问题。历史上第一个关于平面映射的重要结果是关于保向同胚的。平面的所有同胚不是方向保持就是方向反转。在这个提议中，我们将引入一类新的面向映射。每个定向地图都是正向或负向的。我们期望证明平面的所有开放和单调（固有和固有）映射的组合是有取向的。将平面同胚的非分离不变平面连续体上不动点的存在性的已知结果推广到正定向映射的范畴。我们希望这对所有定向地图都是正确的。我们将建立这些问题与包含不变单连通域边界结构的平面分支覆盖映射的不变连续体结构之间的联系。我们将利用元端技术以及连续统理论和动力系统的结果。

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.