engleski

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of zeta functions of fractal strings, which include perturbations of the Riemann zeta function. The distance zeta function $\zeta_A(s):=\int_{; ; ; ; ; ; A_\delta}; ; ; ; ; ; d(x, A)^{; ; ; ; ; ; s-N}; ; ; ; ; ; \mathrm{; ; ; ; ; ; d}; ; ; ; ; ; x$, where $\d>0$ is fixed and $d(x, A)$ denotes the Euclidean distance from $x$ to $A$, has been introduced by the first author in 2009, extending the definition of the zeta function $\zeta_{; ; ; ; ; ; \mathcal L}; ; ; ; ; ; $ associated with bounded fractal strings $\mathcal L=(\ell_j)_{; ; ; ; ; ; j\ge1}; ; ; ; ; ; $ to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence $D(\zeta_A)$ coincides with $D:=\overline\dim_BA$, the upper box (or Minkowski) dimension of~$A$. The (visible) complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of $A$ to a suitable connected neighborhood of the ``critical line'' $\{; ; ; ; ; ; \re s=D\}; ; ; ; ; ; $. (There are none with real part $>D$, since the fractal zeta function is holomorphic on the open right half-plane $\{; ; ; ; ; ; \re s>D\}; ; ; ; ; ; $.) We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $t\to0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{; ; ; ; ; ; N-D}; ; ; ; ; ; (\mathcal M+O(t^\c))$ as $t\to0^+$, with $\c>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{; ; ; ; ; ; N-D}; ; ; ; ; ; (G(\log t^{; ; ; ; ; ; -1}; ; ; ; ; ; )+O(t^\c))$ as $t\to0^+$, where $G$ is a nonconstant periodic function and $\c>0$. In both cases, we show that $\zeta_A$ can be meromorphically extended (at least) to the open right half-plane $\{; ; ; ; ; ; \re s>D-\c\}; ; ; ; ; ; $ and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of $\zeta_A$ evaluated at $s=D$ is shown to be equal to $\mathcal M$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Analogous results are obtained for a class of closely related tube zeta functions, as well as for perturbations of the Riemann zeta function. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line $\{; ; ; ; ; ; \re s=D\}; ; ; ; ; ; $. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct ``maximally-hyperfractal'' compact subsets of $\eR^N$, for $N\ge1$ arbitrary. These are compact subsets of $\eR^N$ such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line $\{; ; ; ; ; ; \re s=D\}; ; ; ; ; ; $.

zeta function ; distance zeta function ; tube zeta function ; fractal set ; fractal drum ; box dimension ; principal complex dimensions ; Minkowski content ; Minkowski measurable set ; residue ; Dirichlet series ; Dirichlet integral ; meromorphic extension ; Sierpi\'nski carpet ; $n$-th order Cantor set ; generalized Cantor set ; hyperfractal

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