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

Abstract

Discrete scale invariance (DSI) has recently been documented in time-to-failure rupture, earthquake processes and financial crashes, in the fractal geometry of growth processes and in 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 new 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.