Characterization of Fat Fractals in Nonlinear Dynamical Systems

Abstract

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in the study of nonlinear dynamical systems. Since fat fractals are not well characterized by their (integer) dimensions, several scaling exponents have been proposed to characterize these sets, and there has been some debate over which exponent is best. We shed light on this issue by examining these exponents and finding relationships between them. Finally, these results lead naturally to quantitative definitions of the notions of sensitivity to parameters and final-state sensitivity.