| In this paper,we study the Assouad dimension of generalized digit restriction sets and the topological Hausdorff dimension of fractal squares.Let b≥2 be an integer,Ib={0,1,…,b-1},D1,D2(?)Ib with D1∩D2 =(?),S1,S2(?)N with S1∩S2＝(?).The set E:={∑k∈Nxk/bk:xk∈D1 for k∈S1;xk ∈ D2 for k ∈ S2;xk ∈ Ib for k ∈N\(S1∪S2)} is called a generalized digit restriction set of data b,D1,D2,S1,S2.As a main result,we obtain the Assouad and the lower Assouad dimension formulas for generalized digit restriction sets.Let n ≥2 and D ={d1,d2,…,dm}(?){0,1,…,n-1}2.Then there is a unique nonempty compact set F satisfying the equation F =1/n(F+D).The set F is called a fractal square of data n and D.As another main result,we compute the topological Hausdorff dimension of fractal squares with n = 3,m = 7.The paper consists of four chapters.Chapter 1 is the background.In Chapter 2,we give and prove the Assouad and the lower Assouad dimension formulas of generalized digit restriction sets.In Chapter 3,we prove that the topological Hausdorff dimension of a fractal square with n = 3,m= 7 is 1+log 2/log 3 or is between 1 and 1 + log 2/log 3.Chapter 4 concludes some further questions. |
