Modulus of continuity for spectral measures of suspension flows over Salem type substitutions

Abstract

We study the spectrum of the self-similar suspension flows of sub-shifts arising from primitive substitutions. We focus on the case where the substitution matrix has a Salem number α as dominant eigenvalue. We obtain a Hölder exponent for the spectral measures for points away from zero and belonging to the field Q(α). This exponent depends only on three parameters of each of these points: its absolute value, the absolute value of its real conjugate and its denominator.