Denjoy's anachronistic topological viewpoint on Aubry transition

The Aubry transition is a phase transition between two types of incommensurate states, originally described as a transition by "breaking of analyticity." Here, we present Denjoy's (anachronistic) viewpoint, who almost 100 years ago described certain mathematical properties of circle homeomorphisms with irrational rotation numbers. The connection between the two lies in the existence of a change of variables from the incommensurate ground state variables to new simple phase variables that rotate by a constant irrational angle. This confers a cyclic order, an essential property of models with the Aubry transition. Denjoy's description indicates that there are two types of cyclic order, distinguished by the regular or singular nature of the change of variables or, in mathematical terms, by the distinction between topological conjugacy vs semiconjugacy. This allows rephrasing the breaking of analyticity as a breaking of topological conjugacy. We illustrate this description with numerical calculations on the Frenkel-Kontorova model.