The fish has no bones
Let \(\mathcal M\) be the set of Borel probabilities on the unit circle which are invariant with respect to the mapping \(x\mapsto 2x\). It is proved that for each function \(\rho_{\omega}(t):=\cos (2\pi (t-\omega))\), there is exactly one element \(\mu\in {\mathcal M}\) which maximizes \(\smallint\rho_{\omega} d\mu\), and that the support of this measure is contained in a semicircle. In particular, the image of the map \(\mu\mapsto\int\exp\{ 2i\pi t\} d\mu\) is compact and a convex set of the complex plane which does not contain any line segments on its boundary. It is also proved that the maximizing measure is periodic for every \(\omega\) except on a set which has measure zero and Hausdorff dimension zero.
scientific article; zbMATH DE number 1525675