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## Complex dimensions generated by essential singularities

Lapidus, Michel L.; Radunović, Goran; Žubrinić, Darko
Complex dimensions generated by essential singularities // 1147th AMS Meeting Program
Honolulu, Havaji, SAD, 2019. str. 204-204 (pozvano predavanje, međunarodna recenzija, sažetak, znanstveni)

Naslov
Complex dimensions generated by essential singularities

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

Sažeci sa skupova, sažetak, znanstveni

Izvornik
1147th AMS Meeting Program / - , 2019, 204-204

Skup
AMS Spring Central and Western Joint Sectional Meeting

Mjesto i datum

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Fractal zeta function ; essential singularity ; pole ; Minkowski content ; Minkowski dimension ; fractal string ; box dimension

Sažetak
Complex dimension are usually defined as poles of the associated fractal zeta function and provide a far-reaching generalization of the classical notion of the Minkowski dimensions. We explore complex dimensions which arise as essential singularities of geometric zeta functions $\zeta_{; ; \mathcal L}; ;$, associated with bounded fractal strings $\mathcal L$ as well as essential singularities of distance zeta functions associated with compact subsets of $\mathbb{; ; R}; ; ^N$. More precisely, for any three prescribed real numbers $D_{; ; \infty}; ;$, $D_1$ and $D$ in $[0, 1]$, such that $D_{; ; \infty}; ; <D_1\le D$, 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 positive real numbers $\a$ such that $\zeta_{; ; \mathcal L}; ;$ is holomorphic in the right open half-plane $\{; ; \re s>\a\}; ;$, except for possible isolated singularities in this half-plane. We achieve this by defining $\mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings. Furthermore, 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 zhen 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