Universal scaling in circle maps (Q914200)

The claimed objective of the paper is to study dissipative dynamic systems and to apply the results to physical problems, such as the one- dimensional Schrödinger equation with a quasi-periodic step-potential or the quasi-periodic oscillations of Hamiltonian systems (incidentally both nondissipative). In the present paper high-order iterations of the circle map are studied and some particular `universal scaling properties' of the devil's staircase are inferred. The reasoning is based on renormalization groups linked to a Farey tree representation of rational numbers. Concrete data are obtained by numerical computations. The abstract results so obtained throw no `visible light' on any known physical problem.