Naslov
Essential singularities of fractal zeta functions

Autori
Lapidus, Michel L. ; Radunović, Goran ; Žubrinić, Darko

Izvornik
Pure and Applied Functional Analysis (2189-3756) 5 (2020), 5; 1073-1094

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Fractal zeta function ; essential singularity ; complex dimension ; generalized Cantor set ; fractal string ; meromorphic function ; meromorphic continuation ; paramorphic function ; paramorphic continuation ; abscissa of paramorphic continuation ; power series of bounded fractal strings ; geometric zeta function ; distance zeta function ; paraharmonic function

Sažetak
We study the essential singularities of geometric zeta functions $\zeta_{; ; ; \mathcal L}; ; ; $, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_\ty$, $D_1$ and $D$ in $[0, 1]$, such that $D_\ty<D_1\le D$, we construct a bounded fractal string $\mathcal L$ such that $D_{; ; ; \rm par}; ; ; (\zeta_{; ; ; \mathcal L}; ; ; )=D_{; ; ; \ty}; ; ; $, $D_{; ; ; \rm mer}; ; ; (\zeta_{; ; ; \mathcal L}; ; ; )=D_1$ and $D(\zeta_{; ; ; \mathcal L}; ; ; )=D$. Here, $D(\zeta_{; ; ; \mathcal L}; ; ; )$ is the abscissa of absolute convergence of $\zeta_{; ; ; \mathcal L}; ; ; $, $D_{; ; ; \rm mer}; ; ; (\zeta_{; ; ; \mathcal L}; ; ; )$ is the abscissa of meromorphic continuation of $\zeta_{; ; ; \mathcal L}; ; ; $, while $D_{; ; ; \rm par}; ; ; (\zeta_{; ; ; \mathcal L}; ; ; )$ is the infimum of all real numbers $\a$ such that $\zeta_{; ; ; \mathcal L}; ; ; $ is holomorphic in the open right half-plane $\{; ; ; \re s>\a\}; ; ; $, except for possible isolated singularities in this half-plane. Defining $\mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_\ty$ of essential singularities of $\zeta_{; ; ; \mathcal L}; ; ; $, contained in the open right half-plane $\{; ; ; \re s>D_{; ; ; \ty}; ; ; \}; ; ; $, coincides with the vertical line $\{; ; ; \re s=D_{; ; ; \ty}; ; ; \}; ; ; $. We extend this construction to the case of distance zeta functions $\zeta_A$ of compact sets $A$ in $\eR^N$, for any positive integer $N$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA