Ordered orbits of the shift, square roots, and the devil's staircase

An orbit of the shift σ: t ↦ 2 t on the circle = ℝ/ℤ is ordered if and only if it is contained in a semi-circle C μ = [μ, μ+½]. We investigate the ‘devil's staircase’ associating to each μ ε the rotation number ν of the unique minimal closed σ-invariant set contained in C μ ; we present algorithms for μ in terms of ν, and we prove (after Douady) that if ν is irrational then μ is transcendental. We apply some of this analysis to questions concerning the square root map, and mode-locking for families of circle maps, we generalize our algorithms to orbits of the shift having ‘sequences of rotation numbers’, and we conclude with a characterization of all orders of points around realizable by orbits of σ.