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 , 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.

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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