Discrete scale invariance in fractals and multifractal measures

@misc{citeulike:835430, abstract = {The theory of fractals and multifractals and their scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. We derive in this work the equations governing the concepts of discrete scale invariance (DSI) and the associated complex exponents in fractals, in multifractal measures and in joint multifractal measures. The set of complex exponents lie symmetrically with respect to the real axis in a vertical strip in the complex plane. In the lattice case, the set of complex exponents lie on finitely many vertical lines periodically and the DSI leads to the log-periodic corrections to scaling implying a preferred scaling ratio that is proved to be independent of the order of the moments. We perform detailed numerical analysis on 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 are no visible log-periodicity, i.e., no preferred scaling ratio since the set of complex exponents are spread irregularly within the complex plane. A non-lattice fractal or multifractal can be approximated by a sequence of lattice fractals or 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.}, author = {Zhou, W. X. and Sornette, D. }, citeulike-article-id = {835430}, eprint = {cond-mat/0408600}, keywords = {fractals, invariance, scale}, month = {Aug}, posted-at = {2007-02-20 19:05:36}, priority = {0}, title = {Discrete scale invariance in fractals and multifractal measures}, url = {http://arxiv.org/abs/cond-mat/0408600}, year = {2004} }

TY - ELEC ID - citeulike:835430 L3 - citeulike-article-id:835430 TI - Discrete scale invariance in fractals and multifractal measures N2 - The theory of fractals and multifractals and their scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. We derive in this work the equations governing the concepts of discrete scale invariance (DSI) and the associated complex exponents in fractals, in multifractal measures and in joint multifractal measures. The set of complex exponents lie symmetrically with respect to the real axis in a vertical strip in the complex plane. In the lattice case, the set of complex exponents lie on finitely many vertical lines periodically and the DSI leads to the log-periodic corrections to scaling implying a preferred scaling ratio that is proved to be independent of the order of the moments. We perform detailed numerical analysis on 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 are no visible log-periodicity, i.e., no preferred scaling ratio since the set of complex exponents are spread irregularly within the complex plane. A non-lattice fractal or multifractal can be approximated by a sequence of lattice fractals or 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. KW - fractals KW - invariance KW - scale AU - Zhou, WX AU - Sornette, D PY - 2004/Aug/27/ UR - http://arxiv.org/abs/cond-mat/0408600 ER -