engleski

Complex dimensions generated by essential singularities

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 $N$.

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

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