a1 School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS
a2 Mathématique, Bâtiment 425, Université de Paris-Sud, 91405 Orsay, France
An orbit of the shift σ: t 2t on the circle = /ℤ is ordered if and only if it is contained in a semi-circle Cμ = [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 Cμ; 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.
(Received January 25 1993)
(Revised March 03 1993)