MULTIFRACTALITY OF GENERALIZED FIBONACCI PROFILES

Abstract

The multifractal properties of a class of one parameter generalized Fibonacci sequences are studied. This class of recursion relations, which is defined by an infinite set of sequences similar to the original Fibonacci’s one, appears for the first time in the study of the Ising model by the real space Migdal-Kadanoff renormalization group approach. The whole set of numbers generated by these equations, when properly arranged over the interval [0,1], gives the exact local magnetization profile of the Ising model on generalized hierarchical lattices at the critical temperature. This profile has a multifractal structure showing that an infinite set of exponents is required to describe how its singularities are distributed. The F(α)-function for the measure defined by the normalized profile is numerically obtained and analyzed. Each value of α characterizes the set of numbers generated by one sequence with arbitrary initial conditions. The exact lower (αmin) and upper (αmax) bounds of the spectra are analytically calculated. For a particular value of the parameter, the original Fibonacci sequence appears, generating a set of numbers diverging with the αmin exponent.