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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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  • [A] Amann, H. (1990) Ordinary Differential Equations. de Gruyter, Berlin-New York. MR 91e:34001
  • [Bea] Beardon, A. (1991) Iteration of Rational Functions. Springer, New York-Berlin-Heidelberg.
  • [Ben] Benford, F. (1938) The law of anomalous numbers. Proceedings of the American Philosophical Society 78, 551-572.
  • [Ber1] Berger, A. (2001) Chaos and Chance. de Gruyter, Berlin-New York.
  • [Ber2] Berger, A. (2002) Multi-dimensional dynamical systems and Benford's law. Submitted.
  • [BD] Brown, J. and Duncan, R. (1970) Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences. Fibonacci Quarterly 8, 482-486. MR 50:12894
  • [D] Diaconis, P. (1979) The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5, 72-81. MR 54:10178
  • [DT] Drmota, M. and Tichy, R. (1997) Sequences, Discrepancies and Applications. Springer, Berlin-Heidelberg-New York. MR 98j:11057
  • [H1] Hill, T. (1995) Base-invariance implies Benford's Law. Proc Amer. Math. Soc. 123, 887-895. MR 95d:60006
  • [H2] Hill, T. (1996) A statistical derivation of the significant-digit law. Statistical Science 10, 354-363. MR 98a:60021
  • [KH] Katok, A. and Hasselblatt, B. (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge. MR 96c:58055
  • [KN] Kuipers, L. and Niederreiter, H. (1974) Uniform Distribution of Sequences. Wiley, New York. MR 54:7415
  • [L] Loéve, M. (1978) Probability Theory II. Springer, New York. MR 58:31324b
  • [N] Newcomb, S. (1881) Note on the frequency of use of the different digits in natural numbers. Amer. J. Math. 4, 39-40.
  • [R] Raimi, R. (1976) The first digit problem. Amer. Math. Monthly 102, 322-327.
  • [SCD] Snyder, M., Curry, J. and Dougherty, A. (2001) Stochastic aspects of one-dimensional discrete dynamical systems: Benford's law. Physical Review E 64, 1-5.
  • [S] Schatte, P. (1988) On mantissa distributions in computing and Benford's law. J. Information Processing and Cybernetics 24, 443-455. MR 90g:60016
  • [TBL] Tolle, C., Budzien, J. and LaViolette, R. (2000) Do dynamical systems follow Benford's law? Chaos 10, 331-337.
  • [W] Weiss, B. (2001) Private communication.

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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

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