## 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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