Singular substitutions of constant length

Abstract

We consider primitive aperiodic substitutions of constant length q and prove that, in order to have a Lebesgue component in the spectrum of the associated dynamical system, it is necessary that one of the eigenvalues of the substitution matrix equals √ q in absolute value. The proof is based on results of M. Queffélec, combined with estimates of the local dimension of the spectral measure at zero.

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

“…The positivity of the Lyapunov exponents follows from the same arguments as in our previous examples. As mentioned before, it is a necessary criterion [17] for a primitive substitution ̺ of constant length L to have an absolutely continuous component in its dynamical spectrum that the substitution matrix of ̺ must have an eigenvalue of modulus √ L. One can easily check that the substitution matrix M = B(0) of ̺ V has eigenvalues {4, 2, 0, 0}, so it satisfies the criterion. However, ̺ V contains coincidences, and hence has pure point spectrum by Dekking's criterion [22].…”

“…We now know that we have the following necessary criterion for the outward iteration. Let us compare this with the necessary criterion from [17] for constant-length substitutions, which is a criterion on the Lyapunov spectrum of the inward iteration. Here, under the non-degeneracy assumption for det(B(k)), we get a condition from the outward iteration, which applies to primitive inflations in general.…”

Lyapunov exponents for binary substitutions of constant length

“…It also gets computationally more involved when one deals with matrices that generate irreducible algebras because there are no subspaces where one can calculate the exponents directly. The unitarity of the restricted Fourier matrix in the Rudin-Shapiro case obviously does not hold in general; consider for example the substitution in [16] which is the same example used in [13] to show that the √ q−condition is necessary but not sufficient. In fact, one can numerically compute that the outward exponents for the a.e.…”

“…This is an extension of the famous Rudin-Shapiro substitution, which has long been known to satisfy that condition. Recently, Berlinkov and Solomyak have shown in [13] that an absolutely continuous component requires that the substitution matrix M ̺ of such q−letter substitution has an eigenvalue whose modulus is √ q. An example beyond the constant length regime is yet to be found.…”

Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction

“…In the binary non-Pisot system studied below, the situation is more complex because the spectrum is mixed, whence it remains to determine the nature of the continuous part. To the best of our knowledge, the answer is not in the literature, though the absence of absolutely continuous components is certainly expected [21,4,2,12]. In anticipation of future work, we do not present the shortest path to the result, as that would mean to restrict more than necessary to methods that are limited to binary alphabets and to this particular example.…”