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 the proportion of values in with mantissa (base ) less than tends to for all in as , for all integer bases . 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 , 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 , where is with , also follow Benford's law. Besides generalizing many well-known results for sequences such as or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.