Goran Radunović

Fractal zeta functions and complex dimensions of relative fractal drums

AbstractThe 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.

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

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.

Distance and tube zeta functions of fractals and arbitrary compact sets

Abstract Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets A of the N-dimensional Euclidean space R N , for any integer N ≥ 1 . It is defined by the Lebesgue integral ζ A ( s ) = ∫ A δ d ( x , A ) s − N d x , for all s ∈ C with Re s sufficiently large, and we call it the distance zeta function of A. Here, d ( x , A ) denotes the Euclidean distance from x to A and A δ is the δ-neighborhood of A, where δ is a fixed positive real number. We prove that the abscissa of absolute convergence of ζ A is equal to dim ‾ B A , the upper box (or Minkowski) dimension of A. Particular attention is payed to the principal complex dimensions of A, defined as the set of poles of ζ A located on the critical line { Re s = dim ‾ B A } , provided ζ A possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, ζ ˜ A ( s ) = ∫ 0 δ t s − N − 1 | A t | d t , called the tube zeta function of A. Assuming that A is Minkowski measurable, we show that, under some mild conditions, the residue of ζ ˜ A computed at D = dim B ⁡ A (the box dimension of A) is equal to the Minkowski content of A. More generally, without assuming that A is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of A. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Bakers theorem from the theory of transcendental numbers.

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

In 2009, the first author introduced a class of zeta functions, called ‘distance zeta functions’, which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of ‘tube zeta functions’, defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of “maximally hyperfractal” compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of \(\mathbb{R}^{N}\)).

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

Abstract 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 ) : = ∫ A δ d ( x , A ) s − N d x , where δ > 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 ζ L associated with bounded fractal strings L = ( l j ) j ≥ 1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D ( ζ A ) coincides with D : = dim ‾ B A , 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 } . 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 → 0 + , 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 ( M + O ( t γ ) ) as t → 0 + , 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 → 0 + , 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 } and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of ζ A evaluated at s = D is shown to be equal to 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 } . 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 R N , for N ≥ 1 arbitrary. These are compact subsets of R N such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line { Re s = D } .

Fractal analysis of Hopf bifurcation at infinity

Using geometric inversion with respect to the origin we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf-Takens bifurcation at infinity.

Classification of Fractal Sets and Concluding Comments

In this last chapter, we first introduce a refinement of the classification of bounded sets in \( \mathbb{R}^{N} \) which had begun with the well-known distinction between Minkowski nondegenerate and Minkowski degenerate sets. Further distinction will be made by classifying fractals according to the properties of their tube functions and allowing, in particular, more general scaling laws than the standard power laws. We then provide a short historical survey concerning notions pertaining to Minkowski measurability and related topics which play an important role in this work. We conclude the book with a few remarks, a long list of open problems, and propose several directions for future research. The research problems and directions proposed here connect many different aspects of fractal geometry, number theory, complex analysis, functional analysis, harmonic analyis, complex dynamics and conformal dynamics, partial differential equations, mathematical physics, spectral theory and spectral geometry, as well as nonsmooth analysis and geometry.

Applications of Distance and Tube Zeta Functions

In this chapter, we show that some fundamental geometric and number-theoretic properties of fractals can be studied by using their distance and tube zeta functions. This will motivate us, in particular, to introduce several new classes of fractals. Especially interesting among them are the transcendentally quasiperiodic sets, since they can be placed at the crossroad between geometry and number theory. We shall need two deep results from transcendental number theory; namely, the theorem of Gel’fond–Schneider, and its extension due to Baker. In this context, the connecting link between the number theory and the geometry of fractals will be their tube zeta functions. A natural extension of the notion of distance zeta function leads us to introducing a general class of weighted zeta functions. Here, we introduce the space L ∞)(Ω): = ∩ p > 1L p (Ω), called the limit L ∞ -space, from which the weight functions are taken. Intuitively, a given weight function w from the space L ∞)(Ω) may only have very mild singularities, say, of logarithmic type. However, the set of singularities may be large, in the sense that its Hausdorff dimension can be arbitrarily close (and even equal) to N. A typical example is the function w(x) = logd(x, A) which appears under the integral sign when we differentiate the distance zeta function. We illustrate the efficiency of the use of distance zeta functions by computing the upper box dimension of several new classes of geometric objects, including geometric chirps, fractal nests and string chirps. These sets are closely related to bounded spirals and chirps in the plane. We also recall the construction of a class of fractals, called zigzagging fractals, for which the upper and lower box dimensions do not coincide, and show that the associated fractal zeta functions are alternating, in a suitable sense.

Fractal Tube Formulas and Complex Dimensions

In this chapter, we reconstruct information about the geometry of relative fractal drums (and, consequently, compact sets) in \(\mathbb{R}^{N}\) from their associated fractal zeta functions. Roughly speaking, given a relative fractal drum (A, Ω) in \(\mathbb{R}^{N}\) (with N ≥ 1 arbitrary), we derive an asymptotic formula for its relative tube function t ↦ | A t ∩Ω | as t → 0+, expressed as a sum taken over its complex dimensions of the residues of its (suitably modified and meromorphically extended) fractal zeta function. The resulting asymptotic formulas are called fractal tube formulas and are valid either pointwise or distributionally, as well as with or without an error term, depending on the growth properties of the associated fractal zeta functions. We note that these fractal tube formulas are expressed either in terms of the tube zeta function \(\tilde{\zeta }_{A,\varOmega }\) or, more interestingly, in terms of the distance zeta function ζ A, Ω . The results of this chapter generalize to higher dimensions and arbitrary relative fractal drums the corresponding ones obtained previously for fractal strings by the first author and M. van Frankenhuijsen. We illustrate these results by obtaining fractal tube formulas for a number of well-known fractal sets, including the Sierpinski gasket and 3-carpet along with higher-dimensional analogs, a version of the graph of the Cantor function (i.e., of the devil’s staircase), fractal strings, fractal sprays, self-similar sprays and tilings, as well as certain non self-similar fractals, such as fractal nests and unbounded geometric chirps. We also apply these results in an essential way in order to obtain and establish a Minkowski measurability criterion for a large class of relative fractal drums (and, in particular, of bounded sets) in \(\mathbb{R}^{N}\), with N ≥ 1 arbitrary. More specifically, under appropriate hypotheses, a relative fractal drum (and, in particular, a bounded set) in \(\mathbb{R}^{N}\) of (upper) Minkowski dimension D is shown to be Minkowski measurable if and only if its only complex dimension with real part equal to D is D itself, and D is simple. We also discuss the notion of fractality defined in our context as the presence of at least one nonreal complex dimension. We show, in particular, that as is expected and intuitive, (a variant of) the Cantor graph (or devil’s staircase) is “fractal” in our sense, whereas as is well known, it is not “fractal” in Mandelbrots’s sense.

Relative Fractal Drums and Their Complex Dimensions

In this chapter, we introduce the notion of relative fractal drums (or RFDs, in short). They represent a simple and natural extension of two fundamental objects of fractal analysis, simultaneously: that of bounded sets in \(\mathbb{R}^{N}\) (i.e., of fractals) and that of bounded fractal strings (introduced by the first author and Carl Pomerance in the early 1990s). Furthermore, there is a natural way to define their associated Minkowski contents and relative distance as well as tube zeta functions. We stress a new phenomenon exhibited by relative fractal drums: namely, their box dimensions can be negative as well (and even equal to −∞). This can be viewed as a property of their ‘flatness’, since it is related to the loss of the cone property. In short, a relative fractal drum (RFD) consists of an ordered pair (A, Ω), where A is an arbitrary (possibly unbounded) subset of \(\mathbb{R}^{N}\) and Ω is an open subset of \(\mathbb{R}^{N}\) of finite volume and such that Ω ⊆ A δ , for some δ > 0. The corresponding zeta function, either a distance or tube zeta function, is denoted by ζ A, Ω or \(\tilde{\zeta }_{A,\varOmega }\), respectively. We show that ζ A, Ω and \(\tilde{\zeta }_{A,\varOmega }\) are connected via a key functional equation, which implies that their poles (i.e., the complex dimensions of the RFD (A, Ω)) are the same. We also extend to this general setting the main results of Chapters 2 and 3 concerning the holomorphicity and meromorphicity of the fractal zeta functions. We introduce the notion of transcendentally quasiperiodic relative fractal drums, using their tube functions. One way of constructing such drums is based on a carefully chosen sequence of generalized Cantor sets, as well as on the use of a classic result by Alan Baker from transcendental number theory. This construction and result extend the corresponding ones obtained in Chapter 3, in which we studied transcendentally quasiperiodic fractal sets. Furthermore, some explicit constructions of RFDs lead us naturally to introduce a new class of fractals, which we call hyperfractals. Particulary noteworthy among them are the maximal hyperfractals, for which the critical line \(\{\mathop{\mathrm{Re}}s = \overline{\dim }_{B}(A,\varOmega )\}\), where \(\overline{\dim }_{B}(A,\varOmega )\) is the relative upper box dimension of (A, Ω) and coincides with the abscissa of convergence of ζ A, Ω or \(\tilde{\zeta }_{A,\varOmega }\), consists solely of nonisolated singularities of the corresponding fractal zeta function (i.e., of the relative distance or tube zeta function), ζ A, Ω or \(\tilde{\zeta }_{A,\varOmega }\).