Transactions of the American Mathematical Society 357(1), pp. 197-219.
ISSN/ISBN: 0002-9947 DOI: 10.1090/S0002-9947-04-03455-5
Abstract: Near a stable fixed point at 0 or ∞, 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), ... , T^{n}(x)} with mantissa (base b) less than t tends to log_{b}t for all t in [1,b) as n → ∞, 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 non-linear systems, exceptional sets may exist. Extensions to non-autonomous dynamical systems are given, and the results are applied to show that many differential equations such as 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.
Bibtex:
@article {MR2098092,
AUTHOR = {Berger, Arno and Bunimovich, Leonid A. and Hill, Theodore P.},
TITLE = {One-dimensional dynamical systems and {B}enford's law},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {357},
YEAR = {2005},
NUMBER = {1},
PAGES = {197--219},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {37A45 (11K06 37A50 37E05 60F05 82B05)},
MRNUMBER = {2098092 (2005m:37017)},
MRREVIEWER = {Peter Raith},
DOI = {10.1090/S0002-9947-04-03455-5},
URL = {http://dx.doi.org/10.1090/S0002-9947-04-03455-5},
}
Reference Type: Journal Article
Subject Area(s): Analysis, Dynamical Systems