Let \(E\) be the self-similar set generated by the iterated function system

\(f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad f_{\beta+1}=\frac{x+\beta+1}{\beta}\)

with \(\beta\ge 3\). Then \(E\) is a self-similar set with complete overlaps, i.e., \(f_{0}\circ f_{\beta+1}={f_{1}\circ f_1}\), but \(E\) is not totally self-similar. We investigate all of its generating iterated function systems, give the spectrum of \(E\), and determine the Hausdorff dimensions and Hausdorff measures of \(E\) and of the sets which contain all points in \(E\) having finite or infinite different codings.

令\（E \）为迭代函数系统生成的自相似集

\（f_0（x）= \ frac {x} {\ beta}，\ quad f_1（x）= \ frac {x + 1} {\ beta}，\ quad f _ {\ beta + 1} = \ frac {x + \ beta + 1} {\ beta} \）

与\（\ beta \ ge 3 \）。那么\（E \）是具有完全重叠的自相似集，即\（f_ {0} \ circ f _ {\ beta + 1} = {f_ {1} \ circ f_1} \），但是\（E \）并非完全自相似。我们研究其所有生成的迭代函数系统，给出\（E \）的谱，并确定\（E \）以及包含\（E \）中所有点具有有限点的集合的Hausdorff尺寸和Hausdorff测度或无限不同的编码。