Some Problems Related To The Self-affine Sets

This dissertation consists of two parts.The first part is related to the generalized multifractal spectrum on a class of self-affine sets,which is arranged at the second to the fourth chapters.And the second part is related to the generalization of the geometrical structure of the general Sierpinski carpets.As a application,we discuss the issue on the intersection of the general Sierpinski carpets with its translations.In chapter 2,we study a class of subsets of the general Sierpinski carpets which are indexed by the group frequencies in the codings.There are two problems,the first is the fact that its natural covering under the self-affine transformation is not a efficient covering,the second is the fact that the limiting frequency of some digits may not exist. However,we obtain its Hausdorff dimension spectrum by density theorem.Especially, we find a class of dense subsets which have the explicit formula of Hausdorff dimension. Meanwhile,we discuss the properties of the corresponding Hausdorff measure. A sufficient condition and a necessary condition for the Hausdorff measures in their dimensions to be positive and finite are given.Furthermore,we prove that the Hausdorff measures in their dimensions are infinite if the conditions fail to hold.In chapter 3,we study a class of subsets of the general Sierpinski carpets which are characterized by insisting that the allowed digits in the expansion occur with propotional group frequencies.By the method of dimension of a measure,we prove that there exists a measure of full Hausdorff dimension supported by these subsets.Thus we obtain its Hausdorff dimension spectrum,which is similar to the variation principle obtained previously in the setting of self-conform.Especially,we give a sufficient condition which Hausdorff dimension is a explicit formula.Finally,A sufficient condition and a necessary condition for the Hausdorff measures in their dimensions to be positive and finite are given.In chapter 4,we study a class of subsets of lailey type self-affine set which are modified by insisting that the allowed digits in the expansions are divided into horizontal fibres and occur with prescribed group frequencies.We prove a version of density theorem by the technique of net measure,and then calculate the Hausdorff and Packing dimensions of this kind of modified fractals and show that they are regular set in the sense of Tricot whereas lalley type self-affine set is even not a regular set. In the second part,which is arranged in chapter 5,we extend the geometrical structure of the general Sierpinski carpets to the so called the multiscale Sierpinski carpets, which is essentially a statistically self-affine sets.There are many ways to make randomly the general Sierpinski carpets.By the method of branching process,Lalley study a version of statistical Sierpinski carpets,which choose randomly rectangles to survive at every step in the geometrical structure of the general Sierpinski carpets,however, must keep the same contractive ratio at all steps.In this paper,we introduce another version of statistical Sierpinski carpets,which permit different contractive ratio at different step,however,the position and numbers of surviving rectangles in the same step are identical.By introducing "mixed" measure,we prove under some condition that there exists a measure of full Hausdorff dimension supported by the multiscale Sierpinski carpets,and thus we obtain its Hausdorff dimension.Packing and Box dimension are also determined.As a application,we discuss the problem of the intersection of the general Sierpinski carpets with its translations,however,the idea comes from that of the intersection of Cantor sets with its translations.