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One-Dimensional Dynamical Systems and Benford's Law
Transactions of the American Mathematical Society
  • Arno Berger, University of Canterbury
  • Leonid A. Bunimovich, Georgia Institute of Technology
  • Theodore P. Hill, Georgia Institute of Technology - Main Campus
Publication Date
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 C2 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.
Citation Information
Arno Berger, Leonid A. Bunimovich and Theodore P. Hill. "One-Dimensional Dynamical Systems and Benford's Law" Transactions of the American Mathematical Society Vol. 357 Iss. 1 (2005) p. 197 - 219
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