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Transactions of the American Mathematical Society

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One-dimensional dynamical systems and Benford's law

Authors: Arno Berger, Leonid A. Bunimovich and Theodore P. Hill
Journal: Trans. Amer. Math. Soc. 357 (2005), 197-219
MSC (2000): Primary 11K06, 37A50, 60A10; Secondary 28D05, 60F05, 70K55
Published electronically: April 16, 2004
MathSciNet review: 2098092
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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.

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Additional Information

Arno Berger
Affiliation: Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Leonid A. Bunimovich
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Theodore P. Hill
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Keywords: Dynamical systems, Benford's law, uniform distribution mod~1, attractor
Received by editor(s): September 15, 2002
Received by editor(s) in revised form: July 10, 2003
Published electronically: April 16, 2004
Additional Notes: The first author was supported by a MAX KADE Postdoctoral Fellowship (at Georgia Tech)
The second author was partially supported by NSF grant DMS-9970215
The third author was partially supported by the Göttingen Academy of Sciences and NSF Grant DMS-9971146
Article copyright: © Copyright 2004 American Mathematical Society