Numerical investigations of discrete scale invariance in fractals and multifractal measures

TL;DR

This paper investigates the numerical properties of lattice and non-lattice multifractals, revealing the origins of scaling regions and proposing methods to extract log-frequencies and approximate non-lattice multifractals.

Contribution

It introduces a novel numerical approach for analyzing lattice multifractals and provides an explicit algorithm to approximate non-lattice multifractals by lattice sequences.

Findings

Identified three different scaling regions in moments of lattice multifractals.

Developed a new numerical method to extract log-frequencies.

Proposed an algorithm to approximate non-lattice multifractals with lattice sequences.

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, {\em{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.