On a kind of self-similar sets with complete overlaps

Let $E$ be the self-similar set generated by the {\it iterated function system} {$$f_0(x)=\frac{x}{\beta },f_1(x)=\frac{x+1}{\beta },f_{\beta +1}=\frac{x+\beta +1}{\beta }$$}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 its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings.