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), T2(x), ... , Tn(x)*} with mantissa (base *b*) less than *t* tends to logbt 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 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 *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.

*Transactions of the American Mathematical Society*Vol. 357 Iss. 1 (2005) p. 197 - 219

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