Frequency locking on the boundary of the barycentre set

Abstract

We consider the doubling map $T:z\mapsto z^2$ of the circle. For each T-invariant probability measure $\mu$ we define its barycentre $b(\mu)=\int_{S^1}z\, d\mu(z)$, which describes its average weight around the circle. We study the set $\Omega$ of all such barycentres, a compact convex set with nonempty interior. Its boundary \box9\ has a countable dense set of points of nondifferentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency locking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit.