Some Problems On Dimensions And Measures Of Self-Affine Sets

In the study of fractal geometry,the measure and dimension are two main re-search areas.In this thesis,we mainly discuss the measure and dimension of self-affine set in R~d.The thesis is divided into four parts.In the first part,we give the basic definitions,theorems and main results of this paper.In the second part,we assume that the self-affine iterated function(IFS) satisfies the weak separation condition(WSC),and study the equivalence of positive Lebesgue measure and non-empty interior for the self-affine set.In this paper,we prove that,if the self-affine set has positive Lebesgue measure,then its interior must be non-empty by discussing the metric density.In the third part,we assume that the IFS satisfies the strong separation condition(SSC),and study the dimension and self-similar measure of the intersection of homogeneous self-similar set and its translation.By using the definition of Hausdorff dimension and the principle of mass distribution,the hausdorff dimension of the inter-section of homogeneous self-similar set and its translation is calculated.Furthermore,a necessary and sufficient condition that the invariant measure of the intersection of the homogeneous self-similar set and its translation is zero has been given.In the fourth part,assume the IFS satisfies SSC,we study the dimension of the intersection of self-affine set and self-translation.By using the pseudo norm ‖·‖_ω and the idea of the third part,the pseudo Hausdorff dimension of the intersection of self-affine set and its translation is given.Then we obtain a range of the Hausdorff dimension.