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

Multi-dimensional dynamical systems and Benford's Law

Arno Berger 1,

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.
Keywords: shadowing., uniform distribution mod 1, Dynamical systems, attractor, Benford's law.
Mathematics Subject Classification: Primary: 11K06, 37A50, 60A10; Secondary: 28D05, 60F05, 70K55.
Citation: Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219
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