Fine structure and complex exponents in power-law distributions from random maps

Discrete scale invariance (DSI) has been suggested recently in time-to-failure rupture, earthquake processes, financial crashes, the fractal geometry of growth processes, and random systems. The main signature of DSI is the presence of log-periodic oscillations correcting the usual power laws, corresponding to complex exponents. Log-periodic structures are important because they reveal the presence of preferred scaling ratios of the underlying physical processes. Here we present evidence of log periodicity overlaying the leading power-law behavior of probability density distributions of affine random maps with parametric noise. The log periodicity is due to intermittent amplifying multiplicative events. We quantify precisely the progressive smoothing of the log-periodic structures as the randomness increases and find a large robustness. Our results provide useful markers for the search of log periodicity in numerical and experimental data.