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

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

Published electronically:
April 16, 2004

MathSciNet review:
2098092

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Near a stable fixed point at 0 or , many real-valued dynamical systems follow Benford's law: under iteration of a map the proportion of values in with mantissa (base ) less than tends to for all in as , for all integer bases . 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 , 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 , where is with , also follow Benford's law. Besides generalizing many well-known results for sequences such as or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.

**[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*, 551-572.**78****[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*, 482-486. MR**8****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.*, 39-40.**4****[R]**Raimi, R. (1976) The first digit problem.*Amer. Math. Monthly*, 322-327.**102****[SCD]**Snyder, M., Curry, J. and Dougherty, A. (2001) Stochastic aspects of one-dimensional discrete dynamical systems: Benford's law.*Physical Review E*, 1-5.**64****[S]**Schatte, P. (1988) On mantissa distributions in computing and Benford's law.*J. Information Processing and Cybernetics*, 443-455. MR**24****90g:60016****[TBL]**Tolle, C., Budzien, J. and LaViolette, R. (2000) Do dynamical systems follow Benford's law?*Chaos*, 331-337.**10****[W]**Weiss, B. (2001) Private communication.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11K06,
37A50,
60A10,
28D05,
60F05,
70K55

Retrieve articles in all journals with MSC (2000): 11K06, 37A50, 60A10, 28D05, 60F05, 70K55

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:
https://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