Multidimensional dynamical systems and Benford’s law. (English) Zbl 1075.37003

It is proved that one-dimensional projections of almost all orbits of many multidimensional dynamical systems follow Benford’s law (\(b\)-Benford sequences \((x_n)\), \(b= 2,3,\dots\), of real numbers are just the sequences \((x_n)\) such that \(\log_b(|x_n|)\) is uniformly distributed modulo 1). It is shown that under (generic) nonresonance conditions on complex \(d\times d\)-matrices \(A\), for every \(x\in\mathbb{C}^{d\times d}\), for every \(x\in \mathbb{C}^d\) real and imaginary part of each nontrivial component of \(O(A,x)= (A^n x)_{n\in\mathbb{N}_0}\) and \((e^{At}z)_{t\geq 0}\) follow Benford’s law. Benford’s laws are also proved for all components of orbits \(O(T, z)\) of more general systems, e.g., for certain linearly dominated systems (for any \(x\) with sufficiently large norm) and certain maps \(T\) with polynomial growth (demonstrating for any component for almost all \(x\) that the orbit \(O(T,x)\) is a \(b\)-Bedford sequence for any \(b= 2,3,\dots\), but exhibiting also dense subsets such that no component of any orbit with sufficiently large norm is a Benford sequence) and for certain complex analytic maps having \(0\) as a stable attracting fixed-point, extending unifying and generalizing also known results obtained, e.g., by number-theoretical methods.


37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
11K06 General theory of distribution modulo \(1\)
60F05 Central limit and other weak theorems
28D05 Measure-preserving transformations


Benford’s law
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