一类分形的构造

The Construction of a Class of Fractals

令f(x)=1,并对0<a<1/2,k≥1定义fa,k(x)=1-2-k+1+akx,则存在R中唯一的非空紧子集Ea满足Ea=f(Ea)∪(∪k≥1fa,k(Ea)).记Ea,n为{fa,k:k≤n}的吸引子,则Ea,n的Hausdorff维数收敛到Ea的Hausdorff维数=-ln2/lna且Ea不是自相似集.

Let f(x)=1, and f a,k (x)=1-2 -k+1 +a kx for 0<a<1/2,k≥1,then there exists a unique nonvoid compact set E a with E a=f(E a)∪(∪ k≥1 f a,k (E a)). Denoted by E a,n the attractor of {f a,k :k≤n} ,the Hausdorff dimension of E a,n converge to that of E a ;moreover,the Hausdorff dimension of E a is -ln 2/ln a and each E a is not self similar.