On Planar Mappings with Invariant Continua

建议：DMS-0072626摘要。这个建议是关于保持某些子连续统不变的平面映射。从历史上看，平面映射的第一个有意义的结果是由保向同胚得到的。平面的同胚不是方向保持的就是方向反转的。在这个建议中，我们将引入一类新的有向映射。每向图要么是正向的，要么是负向的。我们希望证明平面上开的和单调的（真的）映射的所有合成都是定向的。本文将非分离不变平面连续统中同胚不动点存在性的已知结果推广到正定向映射类。我们希望这对所有的定向地图都是正确的。我们将建立这些问题与平面上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。我们将利用素数技巧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.