Rudin-Shapiro Sequence Research Articles

Facteurs des suites de Rudin-Shapiro généralisées

\Integrating paperfolding sequences yields generalized Rudin-Shapiro sequences. We study the factors (subwords) of these sequences, giving an (optimal) property of \half-synchronization and a uniform linear bound for the recurrence function. We also study the powers occurring in these sequences, and we show that the language consisting of all their factors is not contextfree.

Complexité des suites de Rudin-Shapiro généralisées

The complexity of an infinite sequence is defined as the function counting the number of factors of length k in this sequence. We consider the generalized Rudin-Shapiro sequences, which count the number of occurrences of a certain type of blocks in the binary expansion of the nonegative integers, and we prove that their complexity function is an ultimately affine function.

Trace-map invariant and zero-energy states of the tight-binding Rudin-Shapiro model.

The localization of electrons in a one-dimensional tight-binding model with diagonal aperiodicity given by the Rudin-Shapiro sequence is studied in the frame of the transfer-matrix formalism. It is proved that the trace map of the model possesses a polynomial invariant. It is also proved that an infinite sequence of values of the on-site potential exists such that for each of these the energy E=0 lies in the spectrum of the periodic approximants corresponding to the even generations of the chain. Accurate numerical computations show that the states associated to the center of the spectrum are weaker than exponentially localized even for rather small amplitudes of the on-site potential. Scaling laws that govern the spatial decay and self-similarity of these states are derived for various values of the potential strength. It is also shown that the decrease of the potential amplitude qualitatively changes the self-similarity of the wave functions on increasing length scales.

Localization of electrons and electromagnetic waves in a deterministic aperiodic system.

Electron localization and optical transmission in one-dimensional systems with two components distributed according to the Rudin-Shapiro sequence are investigated. The nature of the eigenstates in the diagonal tight-binding model is studied by making use of nonlinear recurrence relations satisfied by the associated transfer matrices and their traces. It is shown that the wave functions display a wide range of features going from weak to exponential localization. Numerical computations lead to the conjecture that the localization property is generic. Nevertheless, a countable dense set of critical values of the on-site potential amplitude is found for which a class of extended states exists. Accordingly, the numerical investigation of the time evolution of the electronic wave packets displays a subdiffusive behavior when the potential amplitude becomes critical. Finally, a study of the optical properties of the Rudin-Shapiro dielectric multilayer shows strong similarities with the behavior of the disordered multilayers regarding the development of gaps in the transmission spectra.

Trace maps of general substitutional sequences.

It is shown that for arbitrary {ital n}, there exists a trace map for any {ital n}-letter substitutional sequence. Trace maps are explicitly obtained for the well-known circle and Rudin-Shapiro sequences which can be defined by means of substitution rules on three and four letters, respectively. The properties of the two trace maps and their consequences for various spectral properties are briefly discussed.

Cantor spectra and scaling of gap widths in deterministic aperiodic systems.

The relationship between geometry and physical properties of aperiodic structures is investigated by considering the example of the tight-binding Schr\"odinger equation in one dimension, where the site potentials are given by an arbitrary deterministic aperiodic sequence. In a perturbative analysis of the integrated density of states, the gaps in the energy spectrum can be ``labeled'' by the singularities of the Fourier transform of the sequence of potentials. This approach confirms known properties of quasiperiodic and almost-periodic systems, and suggests an extension of them to more general sequences, such as those with a singular continuous Fourier transform. There is strong evidence that the spectrum is a Cantor set with zero measure for a much larger class of models than quasiperiodic ones. The dependence of the widths of various gaps on the potential strength is also determined: several different kinds of behavior are obtained, such as a power law with a nontrivial exponent, or an essential singularity. These general results are compared with those of various other approaches for four self-similar sequences generated by substitution, namely the Thue-Morse sequence, the period-doubling sequence, a ``circle sequence,'' and the Rudin-Shapiro sequence.

On sums of Rudin-Shapiro coefficients. II

Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = ∑a(k) and t(n) = ∑(-1)k a(k). In this paper we show that the sequences {s(n)/√n} and {t(n)/√n} do not have cumulative distribution functions, but do have logarithmic distribution functions (given by a specific Lebesgue integral) at each point of the respective intervals [√3/5, √6] and [0, √3]. The functions a(x) and s(x) are also defined for real x ≥ 0, and the function [s(x) – a(x)]/√x is shown to have a Fourier expansion whose coefficients are related to the poles of the Dirichlet series ∑a(n)/n, where Re τ > ½.

Rudin-Shapiro sequences for arbitrary compact groups

AbstractLet G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

Rudin-Shapiro sequences on compact groups

The existence of a certain type of Rudin-Shapiro sequence of functions is shown for all infinite compact groups which are not Lie, thus extending a recent result of the authors to all infinite compact groups.

On nonstrongly regular matrices

Using the Rudin-Shapiro sequence, the existence of a regular but not strongly regular positive matrix that sums { exp ⁡ ( 2 π i k t ) } \{ \exp (2\pi ikt)\} to 0 for all t ∈ ( 0 , 1 ) t \in (0,1) is established. As a corollary it is shown that there exist matrices that sum all almost periodic sequences without possessing the Borel property and vice versa.