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Complex dimensions for IFS with overlaps (2310.08771v3)
Published 12 Oct 2023 in math.DS and math.NT
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 $\zeta(s)=\sum_{n=1}{\infty}d_js$, 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.