Complex dimensions for IFS with overlaps

Abstract

The notion of complex dimension of a one-dimensional Cantor set $C={\bigcap }_{n=1}^{\infty }{C}_n$ dates back decades. It is defined as the set of poles of the meromorphic $\zeta$-function $\zeta (s)={\sum }_{n=1}^{\infty }{d}_j^s$, where $\Re s>0$, 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.

arXiv comment3 pages, no figures