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The singularity spectrum of the fish’s boundary

Published online by Cambridge University Press:  01 February 2008

JULIEN BRÉMONT and
STÉPHANE SEURET
JULIEN BRÉMONT
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, Faculté des Sciences et Technologies, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: bremont@univ-paris12.fr, seuret@univ-paris12.fr)
STÉPHANE SEURET
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, Faculté des Sciences et Technologies, 61 Avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: bremont@univ-paris12.fr, seuret@univ-paris12.fr)
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Abstract

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Let ${\mathcal M}(\T ^1,T)$ be the convex set of Borel probability measures on the circle $\T ^1$ invariant under the action of the transformation $T:x\mapsto 2x$ (mod 1). Its projection on the complex plane by the application $\mu \mapsto \int e^{2i\pi x}\, d\mu (x)$ is a compact convex subset of the unit disc, symmetric with respect to the x-axis, called the ‘fish’ by Bousch. 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 $\infty $, but that every pointwise regularity $\alpha \in [1,\infty ]$ is realized in an uncountable dense set of points. The results rely on fine properties of Sturm measures.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008, pp. 49 - 66
Copyright
Copyright © Cambridge University Press 2008

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