Fixed point theorems in plane continua with applications
Fixed point theorems in plane continua with applications
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DOI：
10.1090/s0065-9266-2012-00671-x
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发表时间：
2010-04
期刊：
影响因子：
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通讯作者：
A. Blokh;R. Fokkink;J. Mayer;L. Oversteegen;E. Tymchatyn
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文献类型：
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作者：
A. Blokh;R. Fokkink;J. Mayer;L. Oversteegen;E. Tymchatyn
我们提出的证明的基本结果，包括哈罗德贝尔开发的，为平面不动点问题：每一个地图的非分离平面连续有一个不动点？其中一些结果已经宣布贝尔早得多，但没有获得的证据。我们定义了简单闭曲线上的映射的变差的概念，并将其与该曲线上的映射的索引联系起来：索引=变差+1。得到了平面上正定向完美映射的一个不动点定理。这推广了Bell在1982年宣布的结果。一个区间到真实的直线的连续映射有一个不动点，该直线的端点向相反的方向移动。我们推广这一点，在平面上的非不变连续映射下，积极面向地图的平面（适当的边界条件）。这些方法意味着在某些情况下，非不变连续统在平面上是退化的。这在复杂动力学中有重要的应用。例如，在一个示例中，我们的结果的一个特殊情况表明，如果X是多项式P的Julia集的非分离不变子连续统，它不包含固定的Cremer点，并且在所有固定点处都不表现出局部旋转，则X必是一个点.由此可见，某些外部射线对多项式Julia集的印象是退化的.
We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval to the real line which sends the endpoints in opposite directions has a fixed point. We generalize this to maps on non-invariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). These methods imply that in some cases non-invariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of our results shows that if $X$ is a non-separating invariant subcontinuum of the Julia set of a polynomial $P$ containing no fixed Cremer points and exhibiting no local rotation at all fixed points, then $X$ must be a point. It follows that impressions of some external rays to polynomial Julia sets are degenerate.