One-dimensional dynamical systems and Benford’s law

Abstract

Near a stable fixed point at 0 or $\infty$, many real-valued dynamical systems follow Benford’s law: under iteration of a map $T$ the proportion of values in $\{x, T(x), T^2(x),\dots , T^n(x)\}$ with mantissa (base $b$) less than $t$ tends to $\log _bt$ for all $t$ in $[1,b)$ as $n\to \infty$, 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 nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as $\dot x=F(x)$, 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.