The Hausdorff Dimension Of Some Fractal Sets

This dissertation focuses on the relation between expansion of a point under different bases,?-dynamical system and the pointwise densities of Cantor measure. We calculate the Hausdorff dimensions of the related fractal sets. The dissertation is divided into six chapters. In the first chapter, we introduce the history of fractal geometry and the backgrounds. After the introductory chapter, we present some preliminaries in the second chapter. Then, we are devoted to disscussing the three issues in details in the following three chapters.In the third chapter, we consider the dimension problem of Furstenberg's conjecture. Furstenberg's conjecture holds for almost all (according to Legesgue measure) real number, because almost all of them are normal to all integer bases. So, a natural question is to ask, besides normal numbers, can one give explicit examples fulfilling the above conjecture? In this dissertation, we construct a Cantor set which makes Fustenberf's conjecture hold. More precisely, we prove that the non-normal number x ?[0,1) which makes the following dimension formula: hold has the full Haudorff dimension.In the fouth chapter, for any real number x ? (0,1], we study the property of the orbit of x1 under the transformation T?. We show that for any given point x0 ? [0,1] and any interval (?0,?1) (?) (1,?o), the set of ? (?)(1,?) such that x0 is not a accumulation point of the orbit of 1 under the transformation T? has the full Hausdorff dimension.In chapter fifth, for a given symmetrical Cantor set, under certain conditions, we obtain the explicit formula of pointwise densities of the Cantor measure on it. More precisely, under the hypothesis the (3-(?))/2<a(?), we obtain the formulas of the upper s-densities of Cantor measure (?)*s(?,x) for every point x ? E(a). Where the notation E(a) denotes a symmetrical Cantor set generated by ?1(x)=ax and ?2(2)= ax+(1-a).In the last chapter, we summarize the main results of this article, and list some relevant topics for further research.