Martine Queffélec

Eigenvalues of Substitution Dynamical Systems

This chapter is devoted to the description of eigenvalues of the dynamical system (X,μ,T) arising from a primitive and aperiodic substitution ζ. We distinguish the case of constant-length substitutions from the case of nonconstant-length ones. In the first case, the result is due to M. Dekking [68] and J.C. Martin [174] who made use of a different approach. The description of the eigenvalues in the general case has been essentially established by B. Host [119]. Another reformulation and a new proof, in somewhat more geometric terms, have been given ten years later [87] and we just outline the ideas. The third part raises the problem of pure point spectrum for substitution dynamical systems, with emphasize on the emblematic Pisot case.

Dynamical Systems Associated with Sequences

In this chapter, we recall elementary facts on topological dynamical systems, and we focus later on a particular case of such systems naturally associated with a bounded sequence. When the sequence takes its values in a finite set, we give an explicit characterization of the dynamical properties of the system. The last section is devoted to the spectral study of sequences with the aid of correlation measures, and to a quick survey on uniform distribution and related sets of integers.

A review of commutative harmonic analysis

This chapter might be skipped at first reading. But we have the feeling that a minimal knowledge of basic facts in harmonic analysis is necessary to understand certain aspects of the analytic theory of Dirichlet series, especially those connected with almost-periodicity, ergodic theory, the Bohr point of view to be developed later, and also universality problems. Therefore, in this introductory chapter, we begin with reminding several basic results of commutative harmonic analysis. Those results, although standard by now, are not so easy to prove, and deserve a careful treatment.

Probabilistic methods for Dirichlet series

The title of this chapter is a little emphatic, because the probabilistic methods will here concentrate essentially about one maximal inequality, which is fairly well-known in harmonic analysis, but will have a specific aspect, due to the Bohr point of view on Dirichlet series. We tried to keep the presentation as self-contained as possible, since the subject may be not completely familiar to some number-theoretists. Let us emphasize that those probabilistic methods have a great flexibility, and are nearly compulsory in some questions, even if the initial proof of the Bohnenblust-Hille theorem, to be proved in the last section, made no use of such methods.

Hardy spaces of Dirichlet Series

The forthcoming spaces \( {{\mathcal{H}}^{p}} \) of Dirichlet series (1 ≤ p ≤ ∞), analogous to the familiar Hardy spaces H p on the unit disk, have been successfully introduced to study completeness problems in Hilbert spaces ([63]), first for p = 2, ∞. Later on, the general case was considered in [10] for the study of composition operators. We will return to that general case further in this chapter, and now concentrate on the cases p = 2, ∞. Here is the initial motivation: let H = L2(0, 1) and ϕ ∈ H, viewed as a 2-periodic odd function on ℝ through its Fourier expansion

Voronin type theorems

\begin{array}{*{20}{c}}{\varphi \left( x \right) = \sum\limits_{n = 1}^\infty {{a_n}\sqrt 2 \sin n\pi x,} } & {\sum\limits_{n = 1}^\infty {{{\left| {{a_n}} \right|}^2} < \infty } } \\\end{array}

Spectral Theory of Unitary Operators

(6.1.1) .

Matrices of Measures

Measure theory, sometimes, brings out the existence of specific elements by giving positive measure to the set of such objects. For want of anything better, it can also be used in the number-theoretical framework to produce classifications of real numbers through their expansions. Ergodic theory will play a role in this purpose.