中文摘要：

席夫定理刻画了开集条件，但不能据此判断一个自相似集是否满足开集条件．作者研究了一类由相似压缩映射S0（x）=x／l，S1（x）=（x＋λ）／l和S2（x）=[x＋（l-1）]／l（l为素数，λ为有理数且λ∈[0，1]）生成的自相似集EA的分形结构与分形雏数，给出了具有完全重叠与不完全重叠两种重叠类型及判定方法，对每种重叠类型给出Hausdorff维数的求法．通过对这类集合的分析发现，即使“简单”的重叠也会产生非常复杂的结构．

英文摘要：

Among the discussion about the measure and dimension of the self-similar sets, the open set condition plays a very important role. The Schief Theorem describes the open set condition, but it cannot determine whether a self-similar set satisfies the open set condition. For the contracting similarities So （x） =x/l, S1（x） = （x ＋ λ）/l and S2（x） = Ix ＋ （l - 1） ]/l（l is a prime number and λ ∈ [0,1 ] ）, the Hausdorff dimension and the structure of EA, an invariant set with respect to S0,S1 ,S2, are studied. Two classifications of complete overlap, no complete overlap, and the corresponding criterions are established. For each classification, the method of getting the Hausdorff dimension is given. It is found that even if “simple” overlap will produce very complex structure by analysis of these sets.