Continuum Theory and Dynamical Systems

连续体理论和动力系统

• 批准号: RGPIN-2014-05725
• 负责人: Tuncali, Murat
• 金额: $ 0.8万
• 依托单位: Nipissing University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611807
• 关键词: Continuum; Theory; Dynamical; Systems

I am interested in problems rising out of general topology, continuum theory and topological dynamics. A continuum means a compact connected Hausdorff space. The Hahn-Mazurkiewicz theorem characterizes locally connected metric continua as continuous images of [0,1]. However, in the nonmetric case, IOK’s (continuous images of ordered continua) are restricted. Mardešic proved that IOK’s are rim-metrizable. M.E. Rudin (2001) characterized the continuous images of compact ordered spaces as the class of monotonically normal compact spaces. Now, there are many intriguing problems concerning IOK’s, rim-metrizable compact spaces and perfectly normal compacta. One is: Characterize rim-metrizable, perfectly normal locally connected continua. This problem is related to the well-known problem of M.E. Rudin: Is it consistent that each perfectly normal locally connected continua is metrizable? There are examples of perfectly normal locally connected continua which are rim-metrizable. Under Continuum Hypothesis or the negation of the Suslin Hypothesis, a variety of perfectly normal spaces can be constructed using various techniques. Banakh, Fedorchuk, Nikiel and I showed that if there are Suslin lines, then there is an example of a nowhere locally connected, nonmetric, Suslinian continuum which is hereditarily separable. We also proved that if there are no Suslin lines, each Suslinian continuum is metrizable. One natural question to consider is whether each hereditarily separable, Suslinian, locally connected continuum is metrizable. One of the areas with which continuum theory intersects widely is Complex Dynamics. One of the interesting problems concerning Julia sets is: Does there exist a rational map whose Julia set is an indecomposable continuum, i.e. a continuum which cannot be written as the union of two proper subcontinua? Devaney and his colleagues obtained examples of Julia sets of exponential maps which are indecomposable. There are results establishing a criterion when a rational Julia set is indecomposable without the existence of buried points. Buried points are the points which do not lie in the boundary of any Fatou component. An intriguing problem is concerned with understanding the topological nature of buried points of a rational Julia set. Banakh and I (2007) studied Hölder maps of [0,1] onto a Peano continuum, a locally connected metric continuum, and introduced the notion of Hölder dimension. Hölder dimension is equal to Fractal dimension for any Peano continuum with a convex metric. A main problem is to characterize Peano continua with Hölder dimension 1/c. For c=1, Fremlin (1992) showed that X has finite length iff X is an image of [0,1] under a Lipschitz map. Eilenberg and Harrold (1943) studied continua of finite length and such continua are characterized as inverse limits of graphs with monotone bonding maps. The problem stated above relates to the problem of characterizing Peano continua of finite volume. Krupski and I studied countable rank maps of continua and obtained interesting results. A map is of countable rank if it has at most countably many nondegenerate fibers. Each continuum of finite length can be obtained as an inverse limit of graphs with countable rank bonding maps. It will be interesting to study inverse limits of Peano continua of dimension >1 with countable rank bonding maps. Bing (1949) proved that a Peano continuum admits a convex metric. Bing asked if the theorem can be generalized to a non-compact space. Nikiel, Stasyuk, Tymchatyn and I proved that if a locally connected, connected metric space has property S, then it admits a convex metric. Now we are interested in improving our result. The techniques we used to construct a convex metric have potential to yield more results.

我对一般拓扑学、连续统理论和拓扑动力学中出现的问题感兴趣。 一个连续统意味着一个紧连通的豪斯多夫空间。Hahn-Mazurkiewicz定理将局部连通度量连续统刻画为[0，1]的连续像。然而，在非度量的情况下，IOK的（连续图像的有序连续）是有限的。Mardešic证明了IOK是可边度量化的。法医Rudin（2001）将紧序空间的连续象刻画为单调正规紧空间类。目前，关于IOK空间、边可度量化紧空间和完全正规紧空间有许多有趣的问题。一个是：刻画边可度量化的、完全正规的局部连通连续统。这个问题与著名的M.E.问题有关。鲁丁：每个完全正规局部连通连续统都是可度量化的，这是一致的吗？有一些完全正规的局部连通连续统是边可度量化的例子。在连续统假设或Suslin假设的否定下，可以使用各种技术构造各种完全正规空间。Banakh，Fedorchuk，Nikiel和我证明了如果有Suslin线，那么就有一个无处局部连通的非度量Suslinian连续统的例子，它是遗传可分的。我们还证明了，如果没有Suslin线，每个Suslinian连续统是可度量化的。一个需要考虑的自然问题是，每个遗传可分的、Suslinian的、局部连通的连续统是否是可度量化的。 与连续统理论广泛交叉的领域之一是复杂动力学。关于Julia集的一个有趣的问题是：是否存在一个有理映射，它的Julia集是一个不可分解的连续统，即一个连续统不能写成两个真子连续统的并集？Devaney和他的同事们得到的例子朱莉娅集的指数映射是不可分解的。有结果建立了一个标准时，合理的朱莉娅集是不可分解的，没有埋点的存在。埋点是指不位于任何Fatou组件边界内的点。一个有趣的问题是关于理解一个合理的Julia集的埋点的拓扑性质。 Banakh和I（2007）研究了[0，1]到Peano连续统（局部连通度量连续统）的Hölder映射，并引入了Hölder维数的概念。对于任何具有凸度量的Peano连续统，Hölder维数等于分形维数。一个主要的问题是用Hölder维数1/c刻画Peano连续统。当c = 1时，Fremlin（1992）证明了X有有限长当且仅当X是[0，1]在Lipschitz映射下的像。Eilenberg和Harrold（1943）研究了有限长的连续统，并将其刻画为具有单调键映射的图的逆极限。上述问题涉及有限体积Peano连续统的特征化问题。Krupski和我研究了连续统的可数秩映射，得到了一些有趣的结果。一个映射是可数秩的，如果它至多有可数个非退化纤维。每个有限长的连续统可以作为具有可数秩键合映射的图的逆极限而得到。研究具有可数秩键映射的维数大于1的Peano连续统的逆极限是一个有趣的问题。 Bing（1949）证明了一个Peano连续统允许一个凸度量。宾问定理是否可以推广到一个非紧空间。Nikiel，Stasyuk，Tymchatyn和我证明了如果一个局部连通的度量空间具有性质S，那么它允许一个凸度量。现在我们有兴趣改善我们的结果。我们用来构造凸度量的技术有可能产生更多的结果。