Two Problems In Fractal Geometry

In this paper, we study two problems of fractal geometry: bilipschitz embeddingsof self-similar sets in a compact metric space, and optimal sets of product of the Cantorset.Given two sets in metric spaces, whether a bilipschitz embedding between themexists, in fact, is the question whether one of them has a subset bilipschitz equivalentwith the other. This is an interesting problem, but even for self-similar sets satisfyingthe strong separation condition, their bilipschitz equivalence is very complicated. Fora self-similar set satisfying the strong separation condition in a compact metric space,we can find a self-similar set satisfying the strong separation condition in Euclideanspace such that they are bilipschitz equivalent. Then we can restrict ambient spaces inEuclidean spaces, for a self-similar set satisfying the strong separation condition.For the topic of bilipschitz embeddings of self-similar sets in a compact metricspace, we show the main results that each self-similar set satisfying the strong separa-tion condition can be bilipschitz embedded into each self-similar set with larger Haus-dorf dimension. A bilipschitz embedding between two self-similar sets satisfying thestrong separation condition with same Hausdorf dimensiion is only possible if the twosets are bilipschitz equivalent. To prove these, we introduce a new definition which iss-structure, and show that some classical fractals, besides of self-similar sets satisfyingthe strong separation condition, have s-structure. And with the help of Ahlfors-Davidset, we establish a bilipshitz mapping from a set having s-structure to a self-similar setwith larger Hausdorf dimension.Computing the Hausdorf measure of C×C is a long standing difcult problem,where C is the classical ternary Cantor set. It is well known that for a self-similar setsatisfying strong separation condition, to calculate the Hausdorf measure is equiva-lent to determine its optimal sets. We studies optimal sets of C×C: their diameters,measures, symmetries and the shapes. For this purpose, we design several devices: the repulsive principle, a bipartite graph G and a W-function. We show that the diameterof B is between1.2993and1.3082. And we get a best estimation of the Hausdorfmeasure of C×C up to now. Two symmetric properties of B are proved. Finally, weshow that the shape of B is very close to a disk. We conjecture that an optimal set canbe a disk.