Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Multi-dimensional dynamical systems and Benford's Law

Pages: 219 - 237, Volume 13, Issue 1, June 2005      doi:10.3934/dcds.2005.13.219

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Arno Berger - Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand (email)

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 $A\in \mathbb C^{d\times d}$, for every $z\in \mathbb C^d$ real and imaginary part of each non-trivial component of $(A^nz)_{n\in \N_0}$ and $(e^{At}z)_{t\ge 0}$ 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.

Keywords:  Dynamical systems, Benford's law, uniform distribution mod 1, attractor, shadowing.
Mathematics Subject Classification:  Primary: 11K06, 37A50, 60A10; Secondary: 28D05, 60F05, 70K55

Received: December 2003;      Revised: November 2004;      Available Online: March 2005.