Lacunarity exponents in chaotic systems

Many physical processes result in very uneven, apparently random, distributions of matter, characterised by fluctuations of the local density varying over orders of magnitude. Examples include the distribution of stars within galaxies, distribution of debris floating on fluids, distribution of human population, and the distribution of small inertial particles in a turbulent flow. Many of these systems can be described by chaotic dynamical systems and the existence of a power-law describing the high-density regions is consistent with the notion that chaotic systems can have fractal invariant measures. However, the distribution of density in the sparse regions can also have a power-law distribution, with an exponent that we refer to as lacunarity exponent, and which is not related to the fractal properties of the system. Here, we discuss a robust mechanism that explains the wide occurrence of these power laws and gives analytical expressions for the lacunarity exponent in some cases of interest, including simple chaotic models and the problem of particles advected by fluid flow.