Journal of Statistical Physics, Vol. 61, Nos. 3/4, 1990. p. 639-665.
The multifractal properties of maps of the circle exhibited in the preceding paper are analyzed from a simplified approach to the renormalization group of Kadanoff. This "second" renormalization group transformation, whose formulation and interpretation are discussed here, acts on the space of one-time-differentiable coordinate changes which associate a map on the critical manifold to the fixed point of the usual renormalization group. While the dependence of the multifractal moments on the starting point can be described statistically, and in particular through universal amplitude ratios as in paper I, it is shown that Fourier analysis is another possible approach. For all multifractal moments, the low-frequency Fourier coefficients have a universal self-similar scaling behavior analogous to that found for the usual spectrum of circle maps. In the case of the first moment, it is demonstrated that the Fourier coefficients are, within constants, equal to the usual spectrum. The relation between amplitude ratios and Fourier coefficients is established and it is demonstrated that the universal values of the ratios come from the universal low-frequency Fourier coefficients.
Since, for the universal ratios arising in the statistical description, the scaling regime is much more easily accessible than for the spectrum, the statistical approach described in paper I should be more convenient for experiments and could become an alternative to the usual spectral description. The universal statistical description of the multifractal moments adopted here is possible because the choice of the a priori probability for the starting point is demonstrated to be irrelevant.