Numerical investigations of discrete scale invariance in fractals and multifractal measures

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 spread 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. ∗ Corresponding author. Address: KPL F 38.2, Kreuzplatz 5, ETH Zurich, CH-8032 Zurich, Switzerland. Phone: +41 44 632 89 17, Fax: +41 44 632 19 14. Email addresses: [email protected] (Wei-Xing Zhou), [email protected] (Didier Sornette). URL: http://www.er.ethz.ch/ (Didier Sornette). 1 This work was partially supported by the NSFC (Grant 70501011) and the Fok Ying Tong Education Foundation (Grant 101086). Preprint submitted to Chaos, Solitons & Fractals 2 February 2008