Sierpinski地毯上的一个Whitney临界集

Whitney's Critical Sets on the Sierpinski Rug

在Ｓｉｅｒｐｉｎｓｋｉ地毯上构造了一个连通集合Ｅ，Ｅ包含１０个压缩比为１／９的压缩函数生成的自相似集，且满足开集条件，它的Ｈａｕｓｄｏｒｆｆ维数为１ｎ１０／１ｎ９；在连通集合Ｅ上构造一个可微函数，利用该函数证明了 Ｅ是一个Ｗｈｉｔｎｅｙ临界集．

A connected set E is constructed on the Sierpinski rug. E is self-similar set resulted from 10 contraction functions with contraction ratio at 1/9 and which meets open set conditions. It's Hausdorff dimension is 1n10/ln9. A differential function is constructed on the connected set E to prove that e is a Whitney's critical set by dividing the differential function into 3 cases.