Grant: Discovery Grants Program - Individual

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. Mardesic 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.