A generalized shadowing lemma is used to study the generation of Benford<br />sequences under non-autonomous iteration of power-like maps T-j : x -><br />alpha(j)x(beta j) (1 - f(j) (x)), with alpha(j), beta(j) > 0 and f(j) is<br />an element of C-1, f(j) (0) = 0, near the fixed point at x = 0. Under<br />mild regularity conditions almost all orbits close to the fixed point<br />asymptotically exhibit Benford's logarithmic mantissa distribution with<br />respect to all bases, provided that the family (T-j) is contracting on<br />average, i.e. lim(n)->infinity n(-1) Sigma(j)(-1)(n)=1 log beta(j) > 0.<br />The technique presented here also applies if the maps are chosen at<br />random, in which case the contraction condition reads Elogo > 0. These<br />results complement, unify and widely extend previous work. Also, they<br />supplement recent empirical observations in experiments with and<br />simulations of deterministic as well as stochastic dynamical systems.
Berger, A. (2005). Benford’s law in power-like dynamical systems. STOCHASTICS AND DYNAMICS, 5(4), 587–607. https://doi.org/10.1142/S0219493705001602