Multi-dimensional Dynamical Systems and Benford’s Law

One-dimensional projections of (at least) almost all orbits of many multidimensional 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 (Az)n∈N0 and (e 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.