Continuum Theory and Dynamical Systems

我对从一般拓扑学、连续体理论和拓扑动力学中产生的问题感兴趣。 连续体是指紧连通的Hausdorff空间。Hahn-Mazurkiewicz定理刻画了局部连通度量连续体是[0，1]的连续像。然而，在非度量情况下，IOK(有序连续体的连续像)是受限的。Mardešic证明了Iok的边缘是可度量的。M.E.Rudin(2001)刻画了紧序空间的连续像是单调正规紧空间类。现在，关于IOK、可边度量化的紧空间和完全正规紧空间有许多有趣的问题。其一是：刻画了完全正规的局部连通连通连续体。这个问题与M.E.Rudin的著名问题有关：每个完全正常的局部连通连续体都是可度量化的，这是一致的吗？有一些完全正常的局部连通连续体的例子，它们是边缘可度量的。在连续统假设或对Suslin假设的否定下，可以使用各种技巧来构造各种完全正规空间。Banakh，Fedorchuk，Nikil和我证明，如果有Suslin线，那么就有一个例子，没有局部连通的，非度量的，Suslinian连续体，它是遗传可分的。我们还证明了如果没有Suslin线，则每个Suslinian连续体都是可度量化的。一个需要考虑的自然问题是，每一个可遗传分离的、苏斯林的、局部相连的连续体是否都是可度量的。 复杂动力学是连续体理论广泛交叉的领域之一。关于Julia集的一个有趣的问题是：是否存在一个有理映射，其Julia集是一个不可分解的连续体，即不能写成两个真次连续体的并的连续体？Devaney和他的同事得到了不可分解的Julia指数映射集的例子。已有结果建立了有理Julia集在不存在埋点的情况下不可分解的一个判据。埋藏点是指不位于任何FATOU分量边界内的点。一个有趣的问题是关于理解有理Julia集的埋点的拓扑性质。 Banakh和我(2007)研究了[0，1]到Peano连续体(局部连通度量连续体)上的Hölder映射，并引入了Hölder维的概念。对于任何具有凸度量的Peano连续体，Hölder维数等于分数维。刻划Hölder维为1/c的Peano连续体是一个主要问题，Fremlin(1992)证明了当c=1时，X有有限长的当且仅当X是[0，1]在Lipschitz映射下的像.Eilenberg和Harroeld(1943)研究了有限长的连续体，这种连续体被刻画为具有单调结合映射的图的逆极限。上述问题涉及有限体积的Peano连续体的刻画问题。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。如果一个映射至多有可数多个非简并纤维，则该映射是可数秩图。有限长的连续体可以作为具有可数秩键映射的图的逆极限得到。用可数秩键映射研究&gt；1维Peano连续体的逆极限将是很有意义的。 Bing(1949)证明了Peano连续体有凸度量。宾问，这个定理是否可以推广到非紧空间。尼基尔、Stasyuk、Tymchat yn和我证明了如果一个局部连通、连通的度量空间具有S性质，则它允许一个凸度量。现在我们对改善我们的结果很感兴趣。我们用来构造凸度量的技术有可能产生更多的结果。

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.