Dimensions Of Slices Of A Class Of Fractals

In the dissertation, we concern to the intersection of a class of fractals in Rn, which is generated by multi-rules, with (n-m)-dimensional subspace of integral direction vector. Under certain condition, we prove that the Haudorff dimension or box dimension of slices equal to the Hausdorff dimension of the fractals minus m. Other related topics are also discussed.In detail, the fractals we discussed are generated by rules from initial cube. Partition the unit cube [0,1]n into several congruent sub-cubes, and discard several of them according to a given rule which contains the information of positions of several sub-cubes that will be discarded. Apply the same process, to each remaining sub-cubes. The rules we discard sub-cubes in each steps are given. And during a step, the rule we applied to each sub-cubes are the same. Repeat this operation ad infinitum, we get the limit set E, which we called fractals generated by rules. If we use the same rule throughout the whole process, the limit set E is a self-similar set. If we use sequence of rules we used is not eventually periodic, the limit set E is a fractal of Moran structure, called fractals generated by multi-rules.Throughout the dissertation, the rules we used and the orthogonal direction of the sub-space are chosen to satisfy a congruence condition which we called s-star condition.In Chapter3, we discuss intersection of the self-similar fractal E with (n-1)-dimensional hyperplane of integer orthogonal direction and rational intercepts. Under s-star condition we prove that the dimensions equal the dimension of E minus one.In Chapter4, we discuss intersection of the self-similar fractal E with (n-1)-dimensional hyperplane of integer orthogonal direction and irrational intercepts. We prove that s-star condition is a sufficient condition to ensure that the typical Hausdorff dimension of slices takes the value in Marstrand’s theorem, i.e., the dimension of the self-similar set minus one.In Chapter5, the Hausdorff dimension of the intersection of self-similar fractals in Eu-clidean space Rn generated from initial cube pattern with (n-m)-dimensional hyperplane V in a fixed direction is discussed. And we prove that s-star condition is sufficient to ensure that the Hausdorff dimensions of the slices of the fractal sets generated by "multi-rules" take the valve in Marstrand’s theorem, i.e., the dimension of the self-similar sets minus m. For the self-similar fractals generated from with initial cube pattern, this sufficient condition also ensures that the projection measure μV is absolutely continuous with respect to the Lebesgue measure (?)m. When the projection measure is absolutely continuous with respect to the m-dimensional Lebesgue measure, the connection of the local dimension of and the box dimension of slices is also given.