## 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.

### MSC:

 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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