A partial order on \(\times 2\)-invariant measures
Let \(\mathcal M\) be the simplex of \(\times 2-\)invariant Borel probability measures on \(X=[0,1]\), \(b(\mu):=\int x\,d\mu(x)\) the barycentre of the probability measure \(\mu\), \({\mathcal M}_\rho:=\{\mu\in {\mathcal M}: b(\mu)=\rho\}\). Theorem: For every \(\rho\in[0,1]\), the ordered set \(({\mathcal M}_\rho,\prec)\) has a least element. This least element is the Sturmian measure \(S_\rho\) of the rotation number \(\rho.\) The theorem itself is a corollary of a result on \(C^2\) convex functions: let \(f:X\to\mathbb R\) be a \(C^2\) convex function. For every \(\rho\in (0,1) \) there exists a \(\theta \in\mathbb R\) such that the Sturmian measure \(S_\rho\) is a minimizing measure for the function \(f_\theta\) defined by the \(f_\theta(x)= f(x)+\theta x\). A number of corollaries about Sturmian measures in \({\mathcal M}_\rho\) are also given, including facts on the smaller \(f\)-integral, on smallest variance and the result that the Sturmian orbit has largest geometric mean. The result on \(C^2\) convex functions comes partially from a reformulation of the ``Précondition de Sturm'' by \textit{T. Bousch} [Ann. Inst. Henri Poincaré, Probab. Stat. 36, 489--508 (2000; Zbl 0971.37001)]. The paper under review gives a short survey of ergodic optimization and of Sturmian measures as minimizing and maximizing measures for classes of functions.
scientific article; zbMATH DE number 5505927