Suppose compact sets E and F are quasi uniformly disconnected and quasi Ahlfors-David regular.This paper proves that E and F are quasi-Lipschitz equivalent if and only if they have the same Hausdorff dimension.
In this paper, we discuss the Lipschitz equivalence of self-similar sets with triangular pattern. This is a generalization of {1, 3, 5}-{1, 4, 5} problem proposed by David and Semmes. It is proved that if two such self-similar sets are totally disconnected, then they are Lipschitz equivalent if and only if they have the same Hausdorff dimension.
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 "multirules" 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 μVis absolutely continuous with respect to the Lebesgue measure L^m. When μV《 L^m, the connection of the local dimension ofμVand the box dimension of slices is given.
