Benford's law in power-like dynamical systems

Abstract

A generalized shadowing lemma is used to study the generation of Benford sequences under non-autonomous iteration of power-like maps Tj: x αjxβj (1 - fj (x)), with αj βj > 0 and fj ∈ C1, fj (0) = 0, near the fixed point at x = 0. Under mild regularity conditions almost all orbits close to the fixed point asymptotically exhibit Benford's logarithmic mantissa distribution with respect to all bases, provided that the family (Tj) is contracting on average, i.e. limn→∞ n-1 ∑j=1n log βj > 0. The technique presented here also applies if the maps are chosen at random, in which case the contraction condition reads E log β > 0. These results complement, unify and widely extend previous work. Also, they supplement recent empirical observations in experiments with and simulations of deterministic as well as stochastic dynamical systems. © 2005 World Scientific Publishing Company.