Jacques Peyrière

Spectrum and extended states in a harmonic chain with controlled disorder: Effects of the Thue-Morse symmetry

Along the lines of previous work, we give the general framework together with a detailed and rigorous study of the spectrum and Born-von Karman eigenstates of a 1D harmonic chain with controlled disorder determined by the Thue-Morse sequence. The spectrum is a Cantor-like set; we prove numerically that its measure is zero and calculate its Bouligand-Minkowski dimension (box dimension). We prove that the value of the IDS on each of the gaps is (2k+1)/(3·2p), withk andp integers. Finally, we also prove that points in a dense subset of the spectrum give rise to extended states, an exceptional property due to the symmetry of the Thue-Morse substitution which can have important applications to multilayered structures, and we illustrate this situation.

Recurrence of the eigenstates of a Schrödinger operator with automatic potential

We consider the Schrödinger eigenvalue problem in the discrete case with a potential assuming two values distributed according to the automatic sequence of Prouhet-Thue-Morse. We show that there are no localized states and that the generalized eigenvectors are recurrent on a geometrical set stemming from the hierarchical nature of the potential.

Beyond Bragg mirrors: the design of aperiodic omnidirectional multilayer reflectors

A new method that enables omnidirectional reflectivity for all polarizations of incident light over a wide selectable range of wavelengths was used to design aperiodic multilayer mirrors. Choosing the materials, and with the desired threshold value of the reflection coefficient also being given, the resulting reflectors work for all polarizations in a predetermined range of incidence angles and wavelengths. They are 1D photonic aperiodic crystals composed of a stack of layers arranged according to a deterministic aperiodic substitutive sequence appropriately determined. In the calculation of the optical multilayer properties with the Rayleigh–Abeles matrices, the use of the computationally efficient building block recurrence or trace–antitrace map techniques avoids numerical instability and strongly reduces computational time. Experimental measurements of a prototype satisfying predetermined requirements fabricated after the sequence generated by σ:(a, b)→(bba, bbba) show excellent agreement.

MULTIFRACTAL FORMALISMS：BOXED VERSUS CENTERED INTERVALS

There are mainly two approaches to the multifractal analysis of measures. The first one, which is used in applications and in studying problems arising from dynamical systems, uses a hierarchy of boxes. The second one, which is more satisfactory from the viewpoint of geometric measure theory, uses more intrinsic concepts. This article is an account of a work by J. Barral, F. Ben Nasr, and J. Peyriére [3] which provides a bridge between these two theories.

Endomorphismes de certaines algèbres à identités polynomiales

Horowitz [2] defined a mapping from a free group Γ of rank n into a quotient ring R of a ring of polynomials on Z with 2n−1 variables in such a way that the value of this mapping (a Fricke character) at a point w in Γ allows to compute the value at w of the character of any representation of Γ in SL(2,C). He also showed [3] that an endomorphism of Γ induces an endomorphism of R. A way of computing these Fricke characters is to consider a PI-algebra A universal for the situation Γ→SL(2,C) and perform computations in it. Any endomorphism of Γ induces an endomorphism of A. In this work, we show that, in the case of rank 2, some properties of endomorphisms of A coming from endomorphisms of Γ extend to all endomorphisms of A.

The factor composition matrix of sequences

Let S = {a, b} be a two-letter alphabet and s a sequence over S. An infinite matrix (ti,j)i,j ≥ 0 is associated with s in the following way: ti,j = 1 if s has a factor which contains i times a and j times b; otherwise ti,j = 0. This matrix will be called the factor composition matrix (FCM) of the sequence s. In this paper, combinatorial properties of certain sequences are studied via their FCMs. In particular (i) the FCM of the Thue-Morse sequence is shown to be pentadiagonal, and substitutions whose fixed points have the same FCM as the Thue-Morse sequence are determined; (ii) an algorithm for computing the FCM of a sturmian sequence is presented; (iii) the FCMs of fixed points of invertible substitutions are characterized in terms of their singular decompositions.