FREQUENCY LOCKING ON THE BOUNDARY OF THE BARYCENTRE SET

OLIVER JENKINSON

Abstract. We consider the doubling map T : z ↦ z^{2} of the circle. For each T -invariant probability measure μ we define its barycentre b(μ) = ∫_{S^{1}} 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 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.