Dimension of slices through fractals with initial cubic pattern

Abstract

In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R n generated from an initial cube pattern with an (n−m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by “multi-rules” take the value in Marstrand’s theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μV is absolutely continuous with respect to the Lebesgue measure L m . When μV ≪ L m , the connection of the local dimension of μV and the box dimension of slices is given.