Probability and Fourier Duality for Affine Iterated Function Systems

Let d be a positive integer, and let μ be a finite measure on ? ^d . In this paper we ask when it is possible to find a subset Λ in ? ^d such that the corresponding complex exponential functions e _λ indexed by Λ are orthogonal and total in L ²(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L ²(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ? ^d ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.     

Iterated function systems, Ruelle operators, and invariant projective measures

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space comes with a finite-to-one endomorphism which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets in of the same cardinality which generate complex Hadamard matrices.

Our harmonic analysis for these iterated function systems (IFS) is based on a Markov process on certain paths. The probabilities are determined by a weight function on . From we define a transition operator acting on functions on , and a corresponding class of continuous -harmonic functions. The properties of the functions in are analyzed, and they determine the spectral theory of . For affine IFSs we establish orthogonal bases in . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in .

     

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].     

Spectral Theory for Discrete Laplacians

We give the spectral representation for a class of selfadjoint discrete graph Laplacians Δ, with Δ depending on a chosen graph G and a conductance function c defined on the edges of G. We show that the spectral representations for Δ fall in two model classes, (1) tree-graphs with N-adic branching laws, and (2) lattice graphs. We show that the spectral theory of the first class may be computed with the use of rank-one perturbations of the real part of the unilateral shift, while the second is analogously built up with the use of the bilateral shift. We further analyze the effect on spectra of the conductance function c: How the spectral representation of Δ depends on c.     

Steps in the Classification of Cohen–Macaulay Modules Over Singularities of Type xt + y3

Let k be an algebraically closed field of characteristic 3 and i, j, t some positive integers such that 1 i < j < t, i + j t. Then there exist a finite number of nonisomorphic indecomposable maximal Cohen–Macaulay modules N over k[[x, y]] /(x^t + y³) such that N / y N is a direct sum of copies of k[[x]] /(xⁱ), k[[x]] /(x^j) and we describe them completely.     

On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries

We study the asymptotic behavior of solutions to the incompressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non-degenerate, meaning not oriented in a single direction.     

The effect of superconducting fluctuations on the resistance of tunneling junction: The existence of ultrathin surface layer in NbN films

The effect of superconducting fluctuations on quasi-particle current of tunneling junction is discussed. The anomalous contribution in tunneling current from interacting across the barrier fluctuations is found. It gives rise to pseudogap minimum in R_d(V). Similar dependence is observed experimentally in NbN-I-Pb junctions. Reasons are given to suggest that there is some ultrathin layer on the surface of NbN films whose properties drastically differ from the bulk one.     

Détection et approximation de défauts non-réguliers

This Note deals with the identifiability of non-smooth defects by boundary measurements. We prove the uniqueness of the detection by two measurements for arbitrary closed sets satisfying quasi-everywhere a conductivity assumption. This assumption is satisfied by a large class of compact sets, including all the sets which can be written as an arbitrary union of continua of positive diameter. The conductivity is a new regularity concept which is related to the thickness of the set and is to be compared to the Wiener regularity. In order to rigorously justify the numerical approach by the finite element method, we provide a stability result without any a priori smoothness assumptions. To cite this article: Z. Belhachmi, D. Bucur, C. R. Acad. Sci. Paris, Ser. I 340 (2005).