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arXiv 1508.04784math-phmath.MP
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分形的复维数与分形 zeta 函数的亚纯延拓

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

Michel L. Lapidus, Goran Radunović, Darko Žubrinić

备注：
30 pages, 2 figures, improved parts of the paper and shortened the paper by reducing background material, to appear in Journal of mathematical analysis and applications in 2017
发表信息：
J. Math. Anal. Appl. 453 (2017), 458-484
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AI总结：

本文研究分形距离、管及分形弦几何 zeta 函数的亚纯延拓，利用管函数渐近展开界定延拓域并联系留数与 Minkowski 容度，还构造了具高阶主复维数和无穷本质奇点的分形弦。

摘要：

我们研究距离 zeta 函数和管 zeta 函数，以及分形弦的几何 zeta 函数的亚纯延拓。距离 zeta 函数 $ζ_A(s):=\int_{A_δ} d(x,A)^{s-N}\mathrm{d}x$（其中固定 $δ>0$，$d(x,A)$ 表示从 $x$ 到 $A$ 的欧氏距离）将与有界分形弦相关的 zeta 函数定义推广到 $\mathbb{R}^N$ 的任意有界子集 $A$。Lebesgue 收敛横坐标 $D(ζ_A)$ 与 $D:=\overline\dim_BA$（即 $A$ 的上盒维数）一致。$A$ 的复维数是 $A$ 的分形 zeta 函数向“临界线”$\{\Re(s)=D\}$ 的某个适当连通邻域作亚纯延拓后的极点。在假定掌握管函数 $|A_t|$（其中 $A_t$ 是 $A$ 的欧氏 $t$-邻域）当 $t\to0^+$ 时渐近展开第二项的适当信息的条件下，我们建立了若干亚纯延拓结果。我们特别关注一类 Minkowski 可测集，满足当 $t\to0^+$ 时 $|A_t|=t^{N-D}(\mathcal M+O(t^γ))$，其中 $γ>0$；以及一类 Minkowski 不可测集，满足当 $t\to0^+$ 时 $|A_t|=t^{N-D}(G(\log t^{-1})+O(t^γ))$，其中 $G$ 是非常值周期函数且 $γ>0$。在两种情形下，我们都证明 $ζ_A$ 可以亚纯延拓（至少）到开右半平面 $\{\Re(s)>D-γ\}$。此外，我们证明，在相差一个乘法常数的意义下，$ζ_A$ 在 $s=D$ 处的留数分别等于 $\mathcal M$（$A$ 的 Minkowski 容度）和 $G$ 的平均值（$A$ 的平均 Minkowski 容度）。而且，我们构造了一类分形弦，其具有任意指定阶数的主复维数，并在临界线 $\{\Re(s)=D\}$ 上具有无穷多个本质奇点。

英文摘要：

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $ζ_A(s):=\int_{A_δ} d(x,A)^{s-N}\mathrm{d}x$, where $δ>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of $\mathbb{R}^N$. The abscissa of Lebesgue convergence $D(ζ_A)$ coincides with $D:=\overline\dim_BA$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the "critical line" $\{\Re(s)=D\}$. 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 $|A_t|=t^{N-D}(\mathcal M+O(t^γ))$ as $t\to0^+$, with $γ>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(\log t^{-1})+O(t^γ))$ as $t\to0^+$, where $G$ is a nonconstant periodic function and $γ>0$. In both cases, we show that $ζ_A$ can be meromorphically extended (at least) to the open right half-plane $\{\Re(s)>D-γ\}$. Furthermore, up to a multiplicative constant, the residue of $ζ_A$ 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. 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 $\{\Re(s)=D\}$.

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