Multifractal Analysis For Sierpinski Carpet And Generalized Sierpinski Carpet

In this thesis, there are two parts. We studied a visibility problem in the first part. More precisely, we considered that "given β∈(0,1/3), is there a ray Lλ with slope λ∈(0,1) which goes through the origin (0,0) such that the intersection of Lλ and set (Cβ×Cβ) only contains the origin." Based on the method of renormalization we obtained some results.We studied that Relationship between multifractal analysis for Sierpinski carpet and generalized Sierpinski carpet in the second part. For any convex quadrangle, we construct a generalized Sierpinski carpet not self-affine, and calculate its Hausdorff dimension by using a bi-lipschitz mapping. And we use the same mapping to obtain a result about multifractal analysis for the generalized Sierpinski carpet.