Relating the various scaling exponents used to characterize fat fractals in nonlinear dynamical systems

Abstract

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in nonlinear dynamical systems. These sets have allowed quantitative analyses of sensitivity to parameters, final-state sensitivity, and quantum chaos, and have played important roles in discussions of nonlinear twist maps, one-dimensional maps with quadratic maxima, Arnol'd tongues in circle maps, and practal basin boundaries. Unfortunately, since fat fractals have nonzero measure, they must have the same (integer) dimension as the underlying space. Therefore, their dimension is insensitive to the fractal structure and does not provide an appropriate means of characterization. Hence, several scaling exponents have been suggested for this purpose, and there has been some debate over which is best. Also, there has been considerable confusion over how these exponents differ and whether they are simply related. We shed light on this issue by examining these exponents and their properties and deriving relationships between them.