Lipschitz Equivalence Of A Class Of General Sierpinski Triangles And Carpets

In this article, we devote to the study of Hausdorff dimension of random fractal, Lips-chitz equivalence of self-similar sets and related topics. We mainly study the four parts as follows:(1) Lipschitz equivalence of a class of general Sierpinski triangles.Letα=(1,0) andβ=(1/2, (?)/2), andΔ={c1α+c2β:c1+c2≤1 and 0≤c1,c2≤1} be an equilateral triangle with lower left corner (0,0) and side 1. Fix an integer n≥2, we letGiven A (?) D, we call EA=∪a∈A(1/nEA+a) a general Sierpinski triangle with initial pattern{1/nΔ+a}a∈A.In this part, we discuss the Lipschitz equivalence of a class of general Sierpinski triangles described as above. This is a generalization of{1,3,5} {1,4.5} problem proposed by David and Semmes in their book:Fractured fractals and broken dreams-Self-similar geometry through metric and measure, printed in 1997. It is proved that if two such general Sierpinski triangles are totally disconnected, then they are Lipschitz equivalent if and only if they have same Hausdorff dimension.(2) Lipschitz equivalence of a class of general Sierpinski carpetsLet n≥2, A (?){0,1,..., n-1}2. We call a general Sierpinski carpets with initial pattern{1/n[0,1]2+a}a∈A.Since the topological structures of general Sierpinski carpets with connected components are more complex than the case of totally disconnected, then the power tool-graph-directed system-fails, which is usually used to deal with the problem of Lipschitz equivalence of dust-like self-similar sets, because there does not exist any graph-directed system with finite vertices that can represent the general Sierpinski carpet with some connected components, so the problem of Lipschitz equivalence related to general Sierpinski carpet with connected component is still an very difficult and open question. In this part, we discuss the Lipschitz equivalence of a class of general Sierpinski carpets in which all connected components with more than one point are line segments. We find a bi-Lipschitz mapping between two link-separated sets with same type by pairing off the basic sets using the indexing by the corresponding symbol space and get a sufficient condition on two general Sierpinski carpets are Lipschitz equivalent. Several examples will be given to illustrate our idea.(3) Hausdorff dimension of a random cut-out set in Rd.In this part, we discuss the Hausdorff dimension of a random cut-out set in Rd by using martingale techniques:the usual procedure is to define a random measure on the random set as the limit of a sequence of random measures associated with the construction, and to use martingales to deduce properties of the limiting measure and obtain a generalization of the results of a random cut-out set on the line.(4) Hausdorff dimension of a class of general Sierpinski carpets.In this part, we show that the general Sierpinski carpets with initial convex quadrilat-eral patterns and triangle patterns and the classical Sierpinski carpet have same Hausdorff dimension by constructing a metric that is distinct to Euclidean metric.