Disorder versus order: Global multifractal relationship between topological entropies and universal convergence rates

Abstract

A multifractal relation between the topological entropies which describe disorder and the universal convergence rates which describe order is found for the first time in one-dimensional chaotic dynamics. There are infinitely many scales in the interval of primitive words and self-similarity in the interval of non-primitive words. After dealing with the singularity of the universal convergence rates at all points of coarse and fine chaos, we obtain the fractal dimension of the curve h-δ(W)-1/∥W∥ to be 1.65 by the fractal interpolation method based on Barnsley's iterated function systems (IFS). The global metric regularity of the disorder versus the order is characterized by the self-similarity of the intervals and its fractal dimension.