The singularity spectrum of the Fish's boundary

Julien Brémont , Stephane Seuret (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

Let M(T1; T) be the convex set of Borel probability measures on the Circle T1 invariant under the action of the transformation T : x 7! 2x mod (1). Its projection on the complex plane by the application 7! R e2i x d (x) is a compact convex of the unit disc, symmetric with respect to the x-axis, called the \Fish" by T. Bousch [3]. Seeing the boundary of the upper half-Fish as a function, we focus on its local regularity. We show that its multifractal spectrum is concentrated at 1, but that every pointwise regularity 2 [1;1] is realized in a non-denumerable and dense set of points. The results rely on ne properties of Sturm measures.

Domaines: Systèmes dynamiques [math.DS]

Julien Brémont, Stephane Seuret. The singularity spectrum of the Fish's boundary. ETDS, 2008, 28 (1), pp.49-66. ⟨hal-00794118⟩