We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpiński carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpiński carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. We also extend this result to higher dimensional cases and provide an effective method on how to draw the associated Hata graph which allows automatical detection by computer. On the other hand, we obtain a characterization of connected GSCs with local cut points. Toward this end, we first deal with connected GSCs with cut points and then return to the local problem. The former is achieved by dividing the collection of GSCs into two parts, i.e., fragile cases and non-fragile cases, and discussing them separately. An easilychecked necessary condition is presented in addition as a byproduct of the above discussion. Finally, we also look into the size of the associated digit sets and possible numbers of cut points of connected GSCs.

Abstract. We construct covering maps from infinite blowups of the n-dimensional Sierpinski gasket SGn to certain compact fractafolds based on SGn. These maps are fractal analogs of the usual covering maps from the line to the circle. The construction extends work of the second author in the case n = 2, but a different method of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.