Abstract. In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary. 1.

To Benoît Mandelbrot, on the occasion of his jubilee. Abstract. We present an overview of a theory of complex dimensions of selfsimilar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several strands to discuss a possible approach to establishing a cohomological interpretation of the complex dimensions. 1.

Abstract. We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log 3 / log2. Consider a smooth compact spin Riemannian manifold M and the Dirac operator ∂M associated with a fixed Riemannian connexion over the spinor bundle S. Let D denote the extension of ∂M to H, the

Abstract. We study countable sums of two dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. At the end we make an explicit computation of the last module for the unit interval in R. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral and the growth of the number of eigenvalues N(Λ) bounded by Λ behaves, such that N(Λ)/Λ is bounded, but without limit for Λ → ∞. 1.

Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry, spectra and dynamics via certain zeta functions and their poles (the complex dimensions) are used in this text as a springboard to define similar tools fit for the study of multifractals such as the binomial measure. The goal of this work is to shine light on new ideas and perspectives rather than to summarize a coherent theory. Progress has been made which connects these new perspectives to and expands upon classical results, leading to a healthy variety of natural and interesting questions for further investigation and elaboration.

Tube formulas (by which we mean an explicit formula for the volume of an (inner) ε-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl’s celebrated results on spectral asymptotics, and the subsequent relation between curvature and spectrum. Additionally, a tube formula contains information about the dimension and measurability of rough sets. In convex geometry, the tube formula of a convex subset of Euclidean space allows for the definition of certain curvature measures. These measures describe the curvature of sets which may be too irregular to support derivatives. In this survey paper, we describe some recent advances in the development of tube formulas for self-similar fractals, and their applications and connections to the other topics mentioned here.

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.

In this paper, we generalize the zeta function for a fractal string (as in [18]) in several directions. We first modify the zeta function to be associated with a sequence of covers instead of the usual definition involving gap lengths. This modified zeta function allows us to define both a multifractal zeta function and a zeta function for higher-dimensional fractal sets. In the multifractal case, the critical exponents of the zeta function ζ(q, s) yield the usual multifractal spectrum of the measure. The presence of complex poles for ζ(q, s) indicate oscillations in the continuous partition function of the measure, and thus give more refined information about the multifractal spectrum of a measure. In the case of a self-similar set in R n, the modified zeta function yields asymptotic information about both the “box ” counting function of the set and the n-dimensional volume of the ɛ-dilation of the set.

: We consider the one-dimensional potential scattering problem on the line when the potential is given by a self-similar fractal measure. We show that the scattering operator admits a Weyl-Berry like expansion in terms of operators, where the second order term has a non trivial scaling which reflects the fractal nature of the potential. 1 Introduction The aim of this paper is to study the one-dimensional potential scattering problem when the interaction term is a self-similar fractal measure. We want to show how the self-similarity of the potential can be retrieven in the scattering operator. First, we will recall some basic facts on the scattering theory for singular Schrodinger operators on the line. Then we introduce the class of self-similar measures and the associated concept of renormalizable measures and operators. Then we establish a "high-frequency" expansion (in a certain meaning) for the scattering operator, where the second order term mirrors the self-similar nature of ...

. Extending results of [HSS] on the construction of Neumann Laplacians of combs with given essential spectrum S = S ae [0; 1), we show that, in addition to the essential spectrum, a bounded sequence of discrete eigenvalues can be prescribed. More generally, we find that certain types of spectra can be constructed precisely, in a bounded interval [0; s]. 1. Introduction. While in many applications one is naturally led to consider elliptic differential operators with Neumann boundary conditions, much more attention has been devoted to the Dirichlet Laplacian than to the Neumann Laplacian. This may be partially due to the fact that the Neumann case is more subtle, and that less tools are available. For example, domain monotonicity (in the sense of quadratic forms) is trivial for the Dirichlet Laplacian, while it is not true in general in the Neumann case (here the one useful monotonic operation is deleting lower-dimensional surfaces from the domain). Similarly, the Feynman-Kac formula i...