One - Dimensional Dynamical Systems and Benford ’ S Law

Near a stable fixed point at 0 or ∞, many real-valued dynamical systems follow Benford's law: under iteration of a map T the proportion of values in {x, T (x), T 2 (x),. .. , T n (x)} with mantissa (base b) less than t tends to log b t for all t in [1, b) as n → ∞, for all integer bases b > 1. In particular , the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford's law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford's distribution occurs for every x, but for essentially non-linear systems, exceptional sets may exist. Extensions to non-autonomous dynamical systems are given, and the results are applied to show that many differential equations such as ˙ x = F (x), where F is C 2 with F (0) = 0 > F (0), also follow Benford's law. Besides generalizing many well-known results for sequences such as (n!) or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.