View Complete Reference

Berger, A (2005)

Multi-dimensional dynamical systems and Benford's law

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 ∈ Cd × d, for every z ∈ Cd real and imaginary part of each non-trivial component of (Anz)n ∈ N and (eAtz)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.

@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 = {}, }

Reference Type: Journal Article

Subject Area(s): Analysis, Dynamical Systems