Frequency locking on the boundary of the barycentre set. (English) Zbl 1106.37303

Jenkinson, Oliver

Exp. Math. 9, No. 2, 309-317 (2000).

Summary: 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 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.

MSC:

37E10	Dynamical systems involving maps of the circle
11K50	Metric theory of continued fractions
37A05	Dynamical aspects of measure-preserving transformations
37E15	Combinatorial dynamics (types of periodic orbits)

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