Onedimensional dynamical systems and Benford's law
Authors:
Arno Berger, Leonid A. Bunimovich and Theodore P. Hill
Journal:
Trans. Amer. Math. Soc. 357 (2005), 197219
MSC (2000):
Primary 11K06, 37A50, 60A10; Secondary 28D05, 60F05, 70K55
Published electronically:
April 16, 2004
MathSciNet review:
2098092
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Near a stable fixed point at 0 or , many realvalued 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 linearlydominated 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 wellknown 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, BerlinNew York. MR 91e:34001
 [Bea]
Beardon, A. (1991) Iteration of Rational Functions. Springer, New YorkBerlinHeidelberg.
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Benford, F. (1938) The law of anomalous numbers. Proceedings of the American Philosophical Society 78, 551572.
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Berger, A. (2001) Chaos and Chance. de Gruyter, BerlinNew York.
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Berger, A. (2002) Multidimensional dynamical systems and Benford's law. Submitted.
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Brown, J. and Duncan, R. (1970) Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences. Fibonacci Quarterly 8, 482486. MR 50:12894
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Diaconis, P. (1979) The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5, 7281. MR 54:10178
 [DT]
Drmota, M. and Tichy, R. (1997) Sequences, Discrepancies and Applications. Springer, BerlinHeidelbergNew York. MR 98j:11057
 [H1]
Hill, T. (1995) Baseinvariance implies Benford's Law. Proc Amer. Math. Soc. 123, 887895. MR 95d:60006
 [H2]
Hill, T. (1996) A statistical derivation of the significantdigit law. Statistical Science 10, 354363. MR 98a:60021
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Katok, A. and Hasselblatt, B. (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge. MR 96c:58055
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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, 3940.
 [R]
Raimi, R. (1976) The first digit problem. Amer. Math. Monthly 102, 322327.
 [SCD]
Snyder, M., Curry, J. and Dougherty, A. (2001) Stochastic aspects of onedimensional discrete dynamical systems: Benford's law. Physical Review E 64, 15.
 [S]
Schatte, P. (1988) On mantissa distributions in computing and Benford's law. J. Information Processing and Cybernetics 24, 443455. MR 90g:60016
 [TBL]
Tolle, C., Budzien, J. and LaViolette, R. (2000) Do dynamical systems follow Benford's law? Chaos 10, 331337.
 [W]
Weiss, B. (2001) Private communication.
 [A]
 Amann, H. (1990) Ordinary Differential Equations. de Gruyter, BerlinNew York. MR 91e:34001
 [Bea]
 Beardon, A. (1991) Iteration of Rational Functions. Springer, New YorkBerlinHeidelberg.
 [Ben]
 Benford, F. (1938) The law of anomalous numbers. Proceedings of the American Philosophical Society 78, 551572.
 [Ber1]
 Berger, A. (2001) Chaos and Chance. de Gruyter, BerlinNew York.
 [Ber2]
 Berger, A. (2002) Multidimensional 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, 482486. MR 50:12894
 [D]
 Diaconis, P. (1979) The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5, 7281. MR 54:10178
 [DT]
 Drmota, M. and Tichy, R. (1997) Sequences, Discrepancies and Applications. Springer, BerlinHeidelbergNew York. MR 98j:11057
 [H1]
 Hill, T. (1995) Baseinvariance implies Benford's Law. Proc Amer. Math. Soc. 123, 887895. MR 95d:60006
 [H2]
 Hill, T. (1996) A statistical derivation of the significantdigit law. Statistical Science 10, 354363. 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, 3940.
 [R]
 Raimi, R. (1976) The first digit problem. Amer. Math. Monthly 102, 322327.
 [SCD]
 Snyder, M., Curry, J. and Dougherty, A. (2001) Stochastic aspects of onedimensional discrete dynamical systems: Benford's law. Physical Review E 64, 15.
 [S]
 Schatte, P. (1988) On mantissa distributions in computing and Benford's law. J. Information Processing and Cybernetics 24, 443455. MR 90g:60016
 [TBL]
 Tolle, C., Budzien, J. and LaViolette, R. (2000) Do dynamical systems follow Benford's law? Chaos 10, 331337.
 [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 303320160
Email:
bunimovh@math.gatech.edu
Theodore P. Hill
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 303320160
Email:
hill@math.gatech.edu
DOI:
http://dx.doi.org/10.1090/S0002994704034555
PII:
S 00029947(04)034555
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 DMS9970215
The third author was partially supported by the Göttingen Academy of Sciences and NSF Grant DMS9971146
Article copyright:
© Copyright 2004 American Mathematical Society
