Discrete and Continuous Dynamical Systems 13(1), pp. 219-237.
ISSN/ISBN: 1078-0947 DOI: 10.3934/dcds.2005.13.219
Abstract: One-dimensional projections of (at least) almost all orbits of many multi-dimensional dynamical systems are shown to follow Benford's law, i.e.\ their (base b) mantissa distribution is asymptotically logarithmic, typically for all bases b. As a generalization and unification of known results it is proved that under a (generic) non-resonance condition on A ∈ C^{d × d}, for every z ∈ C^{d} real and imaginary part of each non-trivial component of (A^{n}z)_{n ∈ N} and (e^{At}z)_{t ≥ 0} follow Benford's law. Also, Benford behavior is found to be ubiquitous for several classes of non-linear maps and differential equations. In particular, emergence of the logarithmic mantissa distribution turns out to be generic for complex analytic maps T with T(0)=0, |T'(0)|<1. The results significantly extend known facts obtained by other, e.g. number-theoretical methods, and also generalize recent findings for one-dimensional systems.
Bibtex:
@article {MR2128801,
AUTHOR = {Berger, Arno},
TITLE = {Multi-dimensional dynamical systems and {B}enford's law},
JOURNAL = {Discrete Contin. Dyn. Syst.},
FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},
VOLUME = {13},
YEAR = {2005},
NUMBER = {1},
PAGES = {219--237},
ISSN = {1078-0947},
MRCLASS = {37A45 (11K36 28D05 37A50 60F05)},
MRNUMBER = {2128801 (2005m:37016)},
MRREVIEWER = {Reinhard Winkler},
DOI = {10.3934/dcds.2005.13.219},
URL = {http://dx.doi.org/10.3934/dcds.2005.13.219},
}
Reference Type: Journal Article
Subject Area(s): Analysis, Dynamical Systems