Benford's law in power-like dynamical systems

  • Berger A
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A generalized shadowing lemma is used to study the generation of Benford
sequences under non-autonomous iteration of power-like maps T-j : x ->
alpha(j)x(beta j) (1 - f(j) (x)), with alpha(j), beta(j) > 0 and f(j) is
an element of C-1, f(j) (0) = 0, near the fixed point at x = 0. Under
mild regularity conditions almost all orbits close to the fixed point
asymptotically exhibit Benford's logarithmic mantissa distribution with
respect to all bases, provided that the family (T-j) is contracting on
average, i.e. lim(n)->infinity n(-1) Sigma(j)(-1)(n)=1 log beta(j) > 0.
The technique presented here also applies if the maps are chosen at
random, in which case the contraction condition reads Elogo > 0. These
results complement, unify and widely extend previous work. Also, they
supplement recent empirical observations in experiments with and
simulations of deterministic as well as stochastic dynamical systems.

Author-supplied keywords

  • dynamical system; shadowing; Benford's law; unifor

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  • A Berger

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