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Register a new userFractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality
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We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,Omega)$ of subsets of the Euclidean space (under some mild assumptions) which generalizes the notion of a (compact) subset and that of a fractal string. By a fractal tube formula for an RFD $(A,Omega)$, we mean an explicit expression for the volume of the $t$-neighborhood of $A$ intersected by $Omega$ as a sum of residues of a suitable meromorphic function (here, a fractal zeta function) over the complex dimensions of the RFD $(A,Omega)$. The complex dimensions of an RFD are defined as the poles of its meromorphically continued fractal zeta function (namely, the distance or the tube zeta function), which generalizes the well-known geometric zeta function for fractal strings. These fractal tube formulas generalize in a significant way to higher dimensions the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen and later on, by the first author, Pearse and Winter in the case of fractal sprays. They are illustrated by several interesting examples. These examples include fractal strings, the Sierpinski gasket and the 3-dimensional carpet, fractal nests and geometric chirps, as well as self-similar fractal sprays. We also propose a new definition of fractality according to which a bounded set (or RFD) is considered to be fractal if it possesses at least one nonreal complex dimension or if its fractal zeta function possesses a natural boundary. This definition, which extends to RFDs and arbitrary bounded subsets of $mathbb{R}^N$ the previous one introduced in the context of fractal strings, is illustrated by the Cantor graph (or devils staircase) RFD, which is shown to be `subcritically fractal.
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The theory of zeta functions of fractal strings has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions, which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of tube zeta functions, defined using the tube function of a fractal set. These three classes of zeta functions, under the name of fractal zeta functions, exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area.
We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their associated fractal zeta functions. Relative fractal drums represent a far-reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings studied extensively by the first author and his collaborators. In fact, the Minkowski measurability criterion established here is a generalization of the corresponding one obtained for fractal strings by the first author and M. van Frankenhuijsen. Similarly as in the case of fractal strings, the criterion established here is formulated in terms of the locations of the principal complex dimensions associated with the relative drum under consideration. These complex dimensions are defined as poles or, more generally, singularities of the corresponding distance (or tube) zeta function. We also reflect on the notion of gauge-Minkowski measurability of RFDs and establish several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples.
In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions, associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the theory of `geometric zeta functions of bounded fractal strings. In this memoir, we introduce the class of `relative fractal drums (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs, extend (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $infty$-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called {em maximal hyperfractals}, such that the critical line of (absolute) convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $mge1$, and even an infinite sequence of essential singularities along the critical line.
We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.
We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $zeta_A(s):=int_{A_delta} d(x,A)^{s-N}mathrm{d}x$, where $delta>0$ 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 $D(zeta_A)$ coincides with $D:=overlinedim_BA$, 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 $tto0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(mathcal M+O(t^gamma))$ as $tto0^+$, with $gamma>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(log t^{-1})+O(t^gamma))$ as $tto0^+$, where $G$ is a nonconstant periodic function and $gamma>0$. In both cases, we show that $zeta_A$ can be meromorphically extended (at least) to the open right half-plane ${Re(s)>D-gamma}$. 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. 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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