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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] Herbert Amann, Ordinary differential equations, de Gruyter Studies in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1990. An introduction to nonlinear analysis; Translated from the German by Gerhard Metzen. MR 1071170
  • [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] J. L. Brown Jr. and R. L. Duncan, Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences, Fibonacci Quart. 8 (1970), no. 5, 482–486. MR 0360444
  • [D] Persi Diaconis, The distribution of leading digits and uniform distribution 𝑚𝑜𝑑 1, Ann. Probability 5 (1977), no. 1, 72–81. MR 0422186
  • [DT] Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics, vol. 1651, Springer-Verlag, Berlin, 1997. MR 1470456
  • [H1] Theodore P. Hill, Base-invariance implies Benford’s law, Proc. Amer. Math. Soc. 123 (1995), no. 3, 887–895. MR 1233974, 10.1090/S0002-9939-1995-1233974-8
  • [H2] Theodore P. Hill, A statistical derivation of the significant-digit law, Statist. Sci. 10 (1995), no. 4, 354–363. MR 1421567
  • [KH] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • [KN] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • [L] Michel Loève, Probability theory. II, 4th ed., Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 46. MR 0651018
  • [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] Peter Schatte, On mantissa distributions in computing and Benford’s law, J. Inform. Process. Cybernet. 24 (1988), no. 9, 443–455 (English, with German and Russian summaries). MR 984516
  • [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
Email: arno.berger@canterbury.ac.nz

Leonid A. Bunimovich
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: bunimovh@math.gatech.edu

Theodore P. Hill
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: hill@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03455-5
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