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Some dynamical properties of Benford sequences. (English) Zbl 1218.11072

For \(b\in \mathbb{N}\setminus \{1\}\), every positive real number \(x\) is written uniquely as \(x=\langle x \rangle_bb^k\) with \(1\leq \langle x\rangle_b<b\), \(k\in \mathbb{Z}\). A sequence \((x_n)\) of real numbers is said to be \(b\)-Benford if \[ \lim_{n\to \infty}\frac{\#\{j<n:\langle |x_j|\rangle_b\leq t\}}n =\log_bt\quad \text{ for}\;t\in [1,b[; \] the sequence \((x_n)\) is said to be Benford if it is \(b\)-Benford for every \(b\in \mathbb{N}\setminus \{1\}\).
Let \((T_n)\) be a sequence of measurable maps of the real or extended real line into itself. Let \(O_T(x)\) denote the sequence generated by \[ x_n=T_n(x_{n-1}),\quad n=1, 2, \ldots, \] subject to the initial condition \(x_0=x\).
For the autonomous case, that is for \(T_n\) independent of \(n\), the author proves that if \(X\) is a Borel subset of the extended real line \(\overline{\mathbb{R}}\) and \(T: X\to X\) preserves a (Borel) probability measure \(\mu\), then \(\mu(\{x\in X:O_T(x) \text{ is Benford}\})=0\).
On the other hand, the author shows that from asymptotic convexity and eventual expansivity for \((T_n)\) it follows that \(O_{\tilde{T}}(x)\) is Benford for a.e. \(x\geq x_1\), where \(\tilde{T}_n\) does not differ too much from \(T_n\). The assumptions are naturally satisfied by a wide variety of examples.

MSC:

11K06 General theory of distribution modulo \(1\)
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B55 Topological dynamics of nonautonomous systems
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