| Lipschitz equivalence is a kernel problem on geometric measure theory and Fractal geometry. This problem is initiated from the works of famous Mathematician K.Falconer [7,8], G. David and S.Semmes[3]. In recent years, the study of this area is very active, and there are many important progresses.We concern the study of Lipschitz equivalence of dusk-like self-similar sets (the so-called Falconer-Marsh problem), which is the center problem of this paper. In this area, there are many techniques such as, algebraic invariants[8], the measure preserving property of Bi-Lipschitz function [25], matchablc condition[15] and the techniques in the ring[28].The first job we do in this paper is that we introduce and study the higher dimensional Frobenius problem. In the classical Frobenius problem, given m rela-tively prime positive integers a1,…, am, find the largest natural number that is not representable as a non-negative integer combination of a1,..., am. Obviously, this depend on the structure of semi-group a1N+…+amN. We replace the integers a1,…,am by the integral vectors X1,…,Xm in the lattice Zs(where we require that they are in the same half-space), and investigate the structure of semi-group X1N+…+Xm N. At first, we introduce the concept of saturated cone, prove the existence of saturated cone, and give the expression of higher dimensional Frobenius problem; Secondly, we introduce directional growth function, and an explicit for-mula is obtained for the directional growth function when the vectors X1,…, Xm are located on the same hyperplane; At last, the sets of vectors define the semi-group are the same. These rigidity results are proved under the assumption that the defining vectors are coplanar.The second job we do in this paper is that we use the previous results to study of Lipschitz equivalence of self-similar sets. For every self-similar sets, we can associate with it a higher dimensional Frobenius problem. At first, we prove that the directional growth function is a Lipschitz invariant; secondly, using the previous rigidity results, we have solved the so-called Falconer-Marsh problem under the assumption that the defining vectors are coplanar.We conjecture that the directional growth function is still an important Lips-chitz invariant under the general setting. For the computation and rigidity of this function, it will be also an important and difficult problem.In addition, we also study the topological structure of a class of self-affine fractal sets. |
