Frequency Locking on the Boundary of the Barycentre Set

We consider the doubling map T : z Z2 of the circle. For each T-invariant probability measure μ we define its barycentre b(μ) = ∫S1 Z dμ(z), which describes its average weight around the circle. We study the set Ω of all such barycentres, a compact convex set with nonempty interior. Its boundary ∂Ω 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.