Spectral measure at zero for self-similar tilings

Jordan Emme (1, 2)

1 AMU - Aix Marseille Université
2 I2M - Institut de Mathématiques de Marseille

The goal of this paper is to study the action of the group of translations over self-similar tilings in the euclidian space $\mathbb{R}^d$. It investigates the behaviour near zero for spectral measures for such dynamical systems. Namely the paper gives a Hölder asymptotic expansion near zero for these spectral measures. It is a generalization to higher dimension of a result by Bufetov and Solomyak who studied self similar-suspension flows for substitutions. The study of such asymptotics mostly involves the understanding of the deviations of some ergodic averages.

Mots clés: Markov operators; spectral measures; ergodic theory; self-similar tilings; Mathematic Subject Classification; 37B50 Multi-dimensional shifts of finite type; tiling dynamics; 37A30 Ergodic theorems; spectral theory

Domaines: Systèmes dynamiques [math.DS]; Théorie spectrale [math.SP]

Jordan Emme. Spectral measure at zero for self-similar tilings. Moscow Mathematical Journal, 2017, 17 (1), pp.35 - 49. ⟨hal-01328019v2⟩