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: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567349
• 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产生的问题很感兴趣。连续体是指紧连的Hausdorff空间。Hahn-Mazurkiewicz定理将局部连通度量连续图刻画为[0,1]的连续像。然而，在非度量的情况下，IOK的（有序连续图像）是有限的。Mardešic证明了IOK是可边缘度量的。M.E. Rudin（2001）将紧有序空间的连续象刻画为一类单调正规紧空间。现在，关于IOK、可环度量紧化空间和完全正规紧化有许多有趣的问题。一个是：描述可环度制的，完全正规的局部连通连续体。这个问题与M.E. Rudin的著名问题有关：每个完全正规的局部连通连续体是可度量的，这是否一致？有一些完全正规的局部连通连续体的例子，它们是可环度制的。在连续统假设或苏斯林假设的否定下，可以使用各种技术构造各种完全正态空间。Banakh Fedorchuk Nikiel和我证明了如果存在苏斯林线，那么就存在一个无处相连的，非度量的苏斯林连续统的例子它是遗传上可分离的。我们还证明了在没有Suslin线的情况下，每个Suslin连续体是可度量的。一个自然要考虑的问题是，是否每一个遗传上可分离的，Suslinian的，局部连接的连续体是可度量的。复杂动力学是连续统理论广泛交叉的领域之一。关于Julia集合的一个有趣的问题是：是否存在一个有理映射，其Julia集合是一个不可分解的连续统，即一个不能写成两个固有子连续统并的连续统？Devaney和他的同事们得到了不可分解的Julia指数映射集的例子。在没有埋藏点的情况下，给出了有理Julia集不可分解的判据。埋藏点是指不位于任何法头分量边界内的点。一个有趣的问题是如何理解有理Julia集合中埋藏点的拓扑性质。Banakh和I（2007）研究了[0,1]在Peano连续统（一个局部连通的度量连续统）上的Hölder映射，并引入了Hölder维数的概念。Hölder维数等于任何具有凸度量的Peano连续体的分形维数。一个主要的问题是用Hölder维数1/c来描述连续皮亚诺。对于c=1, Fremlin（1992）表明，如果X是Lipschitz映射下[0,1]的像，则X具有有限长度。Eilenberg和Harrold（1943）研究了有限长度的连续体，这种连续体被描述为具有单调键合映射的图的逆极限。上述问题涉及有限体积Peano连续体的刻画问题。Krupski和我研究了连续序列的可数秩映射，得到了有趣的结果。如果一个映射最多有可数的非简并纤维，那么它就是可数的。每个有限长度的连续体都可以作为具有可数秩键合映射的图的逆极限来得到。研究具有可数秩键合映射的>1维Peano连续体的逆极限是一个有趣的问题。Bing（1949）证明了Peano连续统允许一个凸度量。Bing问这个定理是否可以推广到非紧空间。Nikiel, Stasyuk， Tymchatyn和我证明了如果一个局部连通连通度量空间具有性质S，那么它就允许一个凸度量。现在我们感兴趣的是改善我们的结果。我们用来构造凸度量的技术有可能产生更多的结果。