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 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.

ER - TY - Generic T1 - Universal Multifractal Properties of Circle Maps From the Point-of-view of Critical Phenomena .1. Phenomenology JF - Journal of Statistical Physics Y1 - 1990 A1 - Fourcade, B. A1 - A.-M. S. Tremblay AB -The strange attractor for maps of the circle at criticality has been shown to be characterized by a remarkable infinite set of exponents. This characterization by an infinite set of exponents has become known as the "multifractal" approach. The present paper reformulates the multifractal properties of the strange attractor in a way more akin to critical phenomena. This new approach allows one to study the universal properties of both the critical point and of its vicinity within the same framework, and it allows universal properties to be extracted from experimental data in a straightforward manner. Obtaining Feigenbaum's scalling function from the experimental data is, by contrast, much more difficult. In addition to the infinite set of exponents, universal amplitude ratios here appear naturally. To study the crossover region near criticality, a "correlation time," which plays a role analogous to the "correlation length" in critical phenomena, is introduced. This new approach is based on the introduction of a joint probability distribution for the positive integer moments of the closest-return distances. This joint probability distribution is physically motivated by the large fluctuations of the multifractal moments with respect to the choice of origin. The joint probability distribution has scaling properties analogous to those of the free energy close to a critical point.

ER - TY - Generic T1 - Universal properties of multifractal moments: Analogies with critical phenomena T2 - Universalities in Condensed Matter Y1 - 1988 A1 - A.-M. S. Tremblay A1 - Fourcade, B. ED - R. Jullien ED - L. Peliti ED - Rammal, R. ED - N. Boccara ER - TY - Generic T1 - ULTRASONIC STUDY OF THE MAGNETOCONDUCTIVITY OF p-InSb AT LOW TEMPERATURES. JF - Journal of physics. C. Solid state physics Y1 - 1987 A1 - Quirion, G. A1 - Poirier, M. A1 - Cheeke, J.D.N. ER - TY - Generic T1 - Unified Approach to Numerical Transfer Matrix Methods for Disordered Systems: Applications to Mixed Crystals and to Elasticity Percolation JF - Journal de physique - Lettres Y1 - 1985 A1 - Lemieux, M.A. A1 - Breton, P. A1 - A.-M. S. Tremblay ER - TY - Generic T1 - Ultrasonic study of thin superfluid helium flims JF - Physics Letters A Y1 - 1982 A1 - Cheeke, J.D.N. A1 - Morisseau, P. A1 - Poirier, M. ER -