Numerical investigations of discrete scale invariance in fractals and multifractal measures

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Abstract

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, i.e., no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly. © 2009 Elsevier B.V. All rights reserved.

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Zhou, W. X., & Sornette, D. (2009). Numerical investigations of discrete scale invariance in fractals and multifractal measures. Physica A: Statistical Mechanics and Its Applications, 388(13), 2623–2639. https://doi.org/10.1016/j.physa.2009.03.023

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