# American Institute of Mathematical Sciences

April  2005, 13(1): 219-237. doi: 10.3934/dcds.2005.13.219

## Multi-dimensional dynamical systems and Benford's Law

 1 Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Received  December 2003 Revised  November 2004 Published  March 2005

One-dimensional projections of (at least) almost all orbits of manymulti-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 unificationof known results it is proved that under a (generic) non-resonance conditionon $A\in \mathbb C^{d\times d}$, for every $z\in \mathbb C^d$ real and imaginary part of each non-trivialcomponent 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 anddifferential equations. In particular, emergence of the logarithmic mantissadistribution 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.
Citation: Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219
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