Michel L. Lapidus

Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals

We establish an analogue of Weyls classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractalK, such as, for example, the Sierpinski gasket. We consider both Dirichlet and Neumann boundary conditions, as well as Laplacians associated with Bernoulli-type (“multifractal”) measures onK. From a physical point of view, we study the density of states for diffusions or for wave propagation in fractal media. More precisely, let ϱ(x) be the number of eigenvalues less thanx. Then we show that ϱ(x) is of the order ofxdS/2 asx→+∞, where the “spectral exponent”dS is computed in terms of the geometric as well as analytic structures ofK. Further, we give an effective condition that guarantees the existence of the limit ofx−dS/2ϱ(x) asx→+∞; this condition is, in some sense, “generic”. In addition, we define in terms of the above “spectral exponents” and calculate explicitly the “spectral dimension” ofK.

Riemann zeros and phase transitions via the spectral operator on fractal strings

The spectral operator was introduced by Lapidus and van Frankenhuijsen (2006 Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings) in their reinterpretation of the earlier work of Lapidus and Maier (1995 J. Lond. Math. Soc. 52 15?34) on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this review, we present the rigorous functional analytic framework given by Herichi and Lapidus (2012) and within which to study the spectral operator. Furthermore, we give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is quasi-invertible (or equivalently, that its truncations are invertible) if and only if the Riemann zeta function ?(s) does not have any zeros on the vertical line Re(s) = c. Hence, it is not invertible in the mid-fractal case when , and it is quasi-invertible everywhere else (i.e. for all c ? (0, 1) with ) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension and c = 1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker?s 75th birthday devoted to ?Applications of zeta functions and other spectral functions in mathematics and physics?.

Spectral and Fractal Geometry: From the Weyl-Berry Conjecture for the Vibrations of Fractal Drums to the Riemann Zeta-Function

Abstract In this paper, we survey some recent results connecting aspects of spectral geometry and fractal geometry.

Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics

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.

SNOWFLAKE HARMONICS AND COMPUTER GRAPHICS: NUMERICAL COMPUTATION OF SPECTRA ON FRACTAL DRUMS

In this work, we study the steady-states vibrations of the “Koch snowflake drum”, both numerically and by means of computer graphics. In particular, we approximate the smallest 50 eigenvalues (or “frequencies” of the drum), along with the corresponding eigenfunctions (called “snowflake harmonics”) of the Dirichlet Laplacian on the Koch snowflake domain. We describe the numerical methods used in the computations, and we display graphical representations of a selected set of the eigenfunctions (as well as of their gradients). In the case of the first harmonic, the graphical results agree with mathematically derived results (by Lapidus and Pang) concerning gradient behavior (“blow up” or “infinite stress” of the membrane) on the boundary and suggest further conjectures regarding the higher eigenfunctions. According to earlier work by the physicist Sapoval and his collaborators, this research may help better understand the formation and “stabilization” of fractal structures (e.g., coastlines, trees and blood vessels) in nature.

A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH APPLICATIONS TO COMPLEX DIMENSIONS

A formula for the interior e-neighbourhood of the classical von Koch snowflake curve is computed in detail. This function of e is shown to match quite closely with earlier predictions from [La-vF1] of what it should be, but is also much more precise. The resulting ‘tube formula’ is expressed in terms of the Fourier coefficients of a suitable nonlinear and periodic analogue of the standard Cantor staircase function and reflects the self-similarity of the Koch curve. As a consequence, the possible complex dimensions of the Koch snowflake are computed explicitly.

Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators

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.

A trace on fractal graphs and the Ihara zeta function

Starting with Iharas work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We de. ne a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.

Eigenfunctions of the Koch snowflake domain

We obtain non-tangential boundary estimates for the Dirichlet eigenfunctions ϕn and their gradients {∇ϕn} for a class of planar domains Ω with fractal boundaries. This class includes the quasidiscs and, in particular, snowflake-type domains. When applied to the case when Ω is the Koch snowflake domain, one of our main results states that {∇ϕ1(ω)} tends to ∞ or 0 as ω approaches certain types of boundary points (where ϕ1 > 0 denotes the ground state eigenfunction of the Dirichlet Laplacian on Ω). More precisely, let Ob (resp., Ac) denote the set of boundary points which are vertices of obtuse (resp., acute) angles in an inner polygonal approximation of the snowflake curve ∂Ω. Then given νεOb (resp., νε Ac), we show that {∇ϕ1(ω)}→∞ (resp., 0) as ω tends to ν in Φ within a cone based at ν. Moreover, we show that blowup of {∇ϕ1} also occurs at all boundary points in a Cantor-type set. These results have physical relevance to the damping of waves by fractal coastlines, as pointed out by Sapovalet al. in their experiments on the “Koch drum”.

Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights

AbstractLet A be a positive self-adjoint elliptic operator of order 2m on a bounded open set Ω ⊂ℝk. We consider the variational eigenvalue problem (P)