Fractal Dimension Fluctuations for Snapshot Attractors of Random Maps

We consider the determination of the information dimension of a fractal snapshot attractor (i.e., the pattern formed by a cloud of orbits at a fixed time) of a random map. It is found that box-counting estimates of the dimension fluctuate from realization to realization of the random process. These fluctuations about the true dimension value are a result of the unavoidable presence of a finite smallest box size ε* used in the dimension estimation. The main result is that the fluctuations are well-described by a Gaussian probability distribution function whose width is proportional to (log1/ε*)^−1/2. Averaging dimension estimates over many realizations (or over time for a single realization) thus yields a means of obtaining a greatly improved estimate of the true dimension value.