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.
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