Darko Žubrinić

Singular sets of Sobolev functions

Abstract We are interested in finding Sobolev functions with “large” singular sets. Given N,k∈ N , 1 R N , such that its upper box dimension is less than N−kp, we construct a Sobolev function u∈ W k,p ( R N ) which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N−kp. To cite this article: D. Žubrinic, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.

Minkowski content and singular integrals

Abstract Assume that lower and upper d-dimensional Minkowski contents of A⊂ R N are different both from 0 and ∞. We show that the function d(x,A)−γ is integrable in a tubular neighbourhood of A if and only if γ

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.

Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension

In a previous paper (21), the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Fractal dimensions in dynamics

This is an invited article for the Encyclopedia of Mathematical Physics, published by Elsevier in Oxford in 2006. We describe some basic methods of fractal analysis in dynamics. A special emphasis is on the computation of Hausdorff and box dimensions. In particular, we present dimension results for the logistic map, Smale horseshoe, Lorenz and Henon attractors, Julia and Mandelbrot sets, spiral trajectories, attractors appearing in infinite-dimensional systems, and Brownian motion trajectories.

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.

Generalized Fresnel integrals and fractal properties of related spirals

Abstract We obtain a new asymptotic expansion of generalized Fresnel integrals x ( t ) = ∫ 0 t cos q ( s ) d s for large t, where q ( s ) ∼ s p when s → ∞ , and p > 1 . The terms of the expansion are defined via a simple iterative algorithm. Using this we show that the box dimension of the related q-clothoid, also called the generalized Euler or Cornu spiral, is equal to d = 2 p / ( 2 p - 1 ) . Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. We also find oscillatory dimension of Fresnel integrals by studying the corresponding chirps.

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 \(\mathbb{R}^{N}\)).

Box dimension of spiral trajectories of some vector fields in ℝ3

We study the behaviour of Minkowski content of bounded sets under bi-Lipschitzian mappings. Applications include Minkowski contents and box dimensions of spirals in ℝ3, dynamical systems, and singular integrals.