Stochastics and Dynamics 5, pp. 587-607.
ISSN/ISBN: 0219-4937 DOI: 10.1142/S0219493705001602
Abstract: A generalized shadowing lemma is used to study the generation of Benford sequences under non-autonomous iteration of power-like maps T_{j}: x → α_{j} x^{βj}(1- f_{j}(x)), with α_{j}, β_{j} > 0 and f_{j} ∈ C^{1}, f_{j} (0) = 0, near the fixed point at x = 0. Under mild regularity conditions almost all orbits close to the fixed point asymptotically exhibit Benfordâ€™s logarithmic mantissa distribution with respect to all bases, provided that the family (T_{j}) is contracting on average, i.e. lim_{n → ∞} n^{-1} ∑_{j=1}^{n} log j > 0.The technique presented here also applies if the maps are chosen at random, in which case the contraction condition reads E log β > 0. These results complement, unify and widely extend previous work. Also, they supplement recent empirical observations in experiments with and simulations of deterministic as well as stochastic dynamical systems.
Bibtex:
@article {MR2185507,
AUTHOR = {Berger, Arno},
TITLE = {Benford's law in power-like dynamical systems},
JOURNAL = {Stoch. Dyn.},
FJOURNAL = {Stochastics and Dynamics},
VOLUME = {5},
YEAR = {2005},
NUMBER = {4},
PAGES = {587--607},
ISSN = {0219-4937},
MRCLASS = {37B55 (11K06 37A50 37E05)},
MRNUMBER = {2185507 (2008i:37029)},
MRREVIEWER = {Peter Raith},
DOI = {10.1142/S0219493705001602},
URL = {http://dx.doi.org/10.1142/S0219493705001602},
}
Reference Type: Journal Article
Subject Area(s): Analysis, Dynamical Systems