Scaling exponents, relations, and order dependence for circle maps

Abstract

Scaling relations describing devil's staircase structures obtained by iterations of orientation-preserving circle maps with minimal slope converging to zero have been investigated. In the limit, the circle map has a zero slope inflection point and the influence of the order at this point on the scaling exponents is analyzed. We study the behavior of the step sizes and the minimal distance to an integer from a (non-zero) cycle point for the super-stable orbit, and in particular we treat the average behavior. This leads to a definition of an ‘average crossover scale exponent’ ν and an ‘average measure exponent’ β , besides the earlier defined ‘average dimension’ D . We find ν ∼- ν, β ∼- β, and D ∼- D. Moreover, β = (1 − D) ν . Finally, as the order increases we observe that hourglass structures evolve in the Arnold tongues.