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One-dimensional dynamical systems and Benford’s law. (English) Zbl 1123.37006
Summary: 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.

MSC:
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11K06 General theory of distribution modulo \(1\)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60A10 Probabilistic measure theory
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
60F05 Central limit and other weak theorems
82B05 Classical equilibrium statistical mechanics (general)
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