Benford’s law in power-like dynamical systems. (English) Zbl 1122.37008

Berger, Arno

Stoch. Dyn. 5, No. 4, 587-607 (2005).

Summary: A generalized shadowing lemma is used to study the generation of Benford sequences under non-autonomous iteration of power-like maps \(T_j : x \mapsto \alpha_jx^{\beta_j} (1 - f_j(x))\), with \( \alpha_j, \beta_j > 0\) and \(f_j \in C^1,\;f_j(0) = 0\), 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. \[ \lim_{n\to\infty}n^{-1} \sum_{j=1}^n\log\beta_j>0 \] . The technique presented here also applies if the maps are chosen at random, in which case the contraction condition reads \(\text{E}\log\beta > 0\). 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.

MSC:

37A50	Dynamical systems and their relations with probability theory and stochastic processes
60A10	Probabilistic measure theory
37B55	Topological dynamics of nonautonomous systems

Keywords:

dynamical system; shadowing; Benford’s law; uniform distribution mod 1

References:

[1]	Aaronson J., Occupation times of sets of infinite measure for ergodic transformations (2002)
[2]	Benford F., Proc. Amer. Philos. Soc. 78 pp 551–
[3]	Berger A., Chaos and Chance (2001) · Zbl 0984.37001
[4]	DOI: 10.3934/dcds.2005.13.219 · Zbl 1075.37003 · doi:10.3934/dcds.2005.13.219
[5]	DOI: 10.1090/S0002-9947-04-03455-5 · Zbl 1123.37006 · doi:10.1090/S0002-9947-04-03455-5
[6]	Boyarski A., Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension (1997)
[7]	Brown J., Fibonacci Quart. 8 pp 482–
[8]	Davenport H., Michigan Math. J. 10 pp 311–
[9]	DOI: 10.1007/978-1-4612-0991-1 · doi:10.1007/978-1-4612-0991-1
[10]	DOI: 10.1214/aop/1176995891 · Zbl 0364.10025 · doi:10.1214/aop/1176995891
[11]	DOI: 10.1016/S0378-4371(01)00497-6 · Zbl 0983.11046 · doi:10.1016/S0378-4371(01)00497-6
[12]	Hill T. P., Statis. Sci. 10 pp 354–
[13]	DOI: 10.1016/S0378-4371(02)00519-8 · Zbl 0994.91017 · doi:10.1016/S0378-4371(02)00519-8
[14]	DOI: 10.1017/CBO9780511809187 · doi:10.1017/CBO9780511809187
[15]	Kuipers L., Uniform Distribution of Sequences (1974) · Zbl 0281.10001
[16]	Ley E., Amer. Statist. 50 pp 311–
[17]	DOI: 10.1007/978-3-642-78324-1 · doi:10.1007/978-3-642-78324-1
[18]	Nigrini M. J., Digital Analysis Using Benford’s Law: Tests and Statistics for Auditors (2000)
[19]	DOI: 10.1007/978-1-4757-3210-8 · doi:10.1007/978-1-4757-3210-8
[20]	Raimi R., Amer. Math. Mon. 102 pp 322–
[21]	Snyder M., Phys. Rev. E 64 pp 1–
[22]	Schatte P., J. Infor. Process. Cyber. 24 pp 443–
[23]	DOI: 10.1063/1.166498 · doi:10.1063/1.166498
[24]	Walters P., An Introduction to Ergodic Theory (1981) · Zbl 0475.28009