engleski

Lapidus zeta functions of fractal sets

A new class of zeta functions has been discovered by M.L.\ Lapidus in Catania in 2009. It can serve as a bridge between the geometric theory of fractal sets and complex analysis. To any nonempty bounded subset $A$ of $\eR^N$ he associated its zeta function $\zeta_A$ defined by $$ \zeta_A(s)=\int_{; ; A_\d}; ; d(x, A)^{; ; s-N}; ; dx. $$ Here $\d$ is a fixed positive number, $A_\d$ is the $\d$-neighbourhood of $A$, $d(x, A)$ is the Euclidean distance from $x$ to $A$, $s$ is the complex variable, and the integral is taken in the sense of Lebesgue. The basic result is that $\zeta_A(s)$ is analytic on the right half-plane $\re s>\ov\dim_BA$, where $\ov\dim_BA$ is the upper box dimension of $A$. Furthermore, the bound is optimal. We indicate some ingredients of the proof of this result and illustrate it with several examples. These zeta functions enable us to extend the notion of complex dimensions for fractal strings, introduced by M.L.\ Lapidus in 1993, to arbitrary fractal sets. Our joint paper with Goran Radunovi\'c in preparation, entitled `Zeta functions of fractal sets in Euclidean spaces', is a continuation of previous studies of M.L.\ Lapidus and his collaborators on fractal strings and their generalizations over the past two decades. (45min lecture by Darko Žubrinić)

Lapidus zeta function; fractal set; fractal string; the upper box dimension; Minkowski content; residue

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