Substitution dynamical systems : algebraic characterization of eigenvalues

Regularities of the Distribution of β-adic van der Corput Sequences

“…By Théorème 5.1 of this article and since the minimal sequence in D is unique, we have S • Γ = Γ • ∆. Therefore the eigenvalues of S are a subset of the eigenvalues of ∆ and, by Proposition 5 of Ferenczi, Mauduit and Nogueira [9], these eigenvalues are of the form exp(2πiy) with y ∈ Q(β), This concludes the proof of Theorem 1.…”

“…Ferenczi, Mauduit and Nogueira [9] gave a more precise description of the set of eigenvalues of ∆ in their Proposition 4, which is too complicated to be cited here.…”

“…For more details on the spectrum of these dynamical systems, see Chapter 7.3 in Pytheas Fogg [17], but note that the result of [9] is cited uncorrectly: According to Theorem 7.3.28 of [17], the eigenvalues of ∆ associated with the trivial coboundary are in exp(2πiZ[β]), but Z[β] should be Q[β] and the condition on the coboundary is unnecessary. Nevertheless, the author considered the coboundary and showed that all reverse β-substitutions τ have only the trivial coboundary, but the proof is rather lengthy and technical and therefore not given in this article.…”

Sequences with constant number of return words

“…This notion was introduced by Durand [5] to give a nice characterization of primitive substitutive sequences. A slightly different notion of return words was used by Ferenczi, Mauduit and Nogueira [8].…”

Episturmian words: a survey

“…Return words play an important role in the study of minimal subshifts in symbolic dynamics; see for instance [45,46,47,53,106]. In the context of episturmian words, such words have recently proven to be a useful tool in the study of quasiperiodicity (see Section 8 for further details).…”