Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

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

Shaun Bulletta1 and Pierrette Sentenaca2

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

Abstract

An orbit of the shift σ: t xs21A6 2t on the circle S0305004100072236_inline1 = xs211D/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 S0305004100072236_inline1 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 S0305004100072236_inline1 realizable by orbits of σ.

(Received January 25 1993)

(Revised March 03 1993)