Complex dimensions for IFS with overlaps

The notion of complex dimension of a one-dimensional Cantor set $C={\bigcap }_{n=1}^{\mathrm{\infty }}{C}_n$ dates back decades. It is defined as the set of poles of the meromorphic $\zeta$-function $\zeta (s)={\sum }_{n=1}^{\mathrm{\infty }}{d}_j^s$, where $\mathrm{\Re }s>0$, and $d_j$ is the length of the $j$th interval in $C_n$. Following the trend, I switch from sets to measures, which will allow me to generalize the construction to iterated function schemes that do not necessarily satisfy the Open Set Condition.