Lipschitz Equivalence And Hausdorff Dimensions Of Univoque Sets

This dissertation is devoted to studying the Lipschitz equivalence of some classes of self-similar sets with exact overlaps,and estimating the Hausdorff dimension of univoque sets of general self-similar sets.In the first part,we firstly consider a class of self-similar sets with exact overlaps.For those self-similar sets,we rewrite each set as a finite union of some disjoint compact sets,which are associated with each other through a graph directed structure.Therefore a sufficient condition of judging the Lipschitz equivalence of these compact sets is given.Finally,by the technique of graph directed structure,we also investigate the Lipschitz equivalence of a class of self-similar sets with overlaps,which are determined by three homogeneous contraction similarity maps.In the final part,our main consideration is the lower bound evaluation of the Hausdorff dimension of univoque sets.For each given iterated function system,we firstly illustrate a selection mechanism of univoque points.Then we can get a subset of the univoque set,which is the invariant set of a finite or countably infinite contraction similarity maps.The Hausdorff dimension of this subset,which is a lower bound of the Hausdorff dimension of the univoque set,can be given by an explicit equation.Moreover,a necessary and sufficient condition of the equality of these two sets' Hausdorff dimension is also introduced.Finally,we get the exact values of the Hausdorff dimensions of some univoque sets by the above method.