On a kind of self-similar sets with complete overlaps

Abstract

Let $E$ be the self-similar set generated by the {\it iterated function system} {\[ f_0(x)=\frac{x}β,\quad f_1(x)=\frac{x+1}β, \quad f_{β+1}=\frac{x+β+1}β \]}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.

17 pages, 1 figure