Scaling in Fat Fractals

Fat fractals are fractals with positive measure and integer fractal dimension. Their dimension is indistinguishable from that of nonfractals, and is inadequate to describe their fractal properties. An alternative approach can be couched in terms of the scaling of the coarse grained measure. For the more familiar “thin” fractals, the resulting scaling exponent reduces to the fractal codimension, but for fat fractals it is independent of the fractal dimension. Numerical experiments on several examples, including the chaotic parameter values of quadratic mappings, the ergodic parameter values of circle maps, and the chaotic orbits of area-preserving maps, show a power law scaling, suggesting that this is a generic form. This paper reviews several possible methods for defining coarse-grained measure and associated fat fractal scaling exponents, reviews previous work on the subject, and discusses problems that deserve further study.