The singularity spectrum of the fish’s boundary

Abstract

Abstract. Let M(T 1 , T ) be the convex set of Borel probability measures on the Circle T 1 invariant under the action of the transformation T : x −→ 2x mod (1). Its projection on the complex plane by the application µ −→ R e 2iπx dµ(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 ∞, but that every pointwise regularity α ∈ [1, ∞] is realized in a uncountable dense set of points. The results rely on fine properties of Sturm measures.

Sturmian maximizing measures for the piecewise-linear cosine family

“…In [4] we also note that, for sufficiently large q, the interval lengths θ max (p/q)−θ min (p/q) are equal to Kq2 q−1 /(2 q − 1) 2 for one of five explicit constants K; asymptotically this agrees with the lengths of the corresponding parameter intervals for the family g θ (see [14,24]). …”