The fish is strictly convex

Let M = M(T-1, x --> 2x) be the set of Borel probabilities on T-1 which are invariant by angle-doubling. We prove that for each function rho(omega)(t) = cos(2 pi(t - omega)), there is exactly one element mu is an element of M which maximises integral rho d mu, and that the support of this measure is contained in a semicircle. In particular, the image of the map mu --> inte gral exp(2i pi t) d mu, which is a compact and convex set of C, is in fact strictly convex; it doesn't contain any line segments on its boundary. We a lso prove that the maximising measure is periodic for every omega except on a set which has measure zero and Hausdorff dimension zero. (C) 2000 Editio ns scientifiques et medicales Elsevier SAS.