A brief survey of Benford's Law in dynamical systems

TL;DR

This survey documents the extensive presence of Benford's law in dynamical systems across autonomous, non-autonomous, discrete, continuous, deterministic, and stochastic settings. It leverages the connection between leading-digit distributions and uniform distribution modulo $1$ of logarithms to establish BL in a wide variety of models, illustrated by canonical examples such as geometric Brownian motion, Newton's method, and the tent map. While many results show BL on sets of full measure, intricate topological phenomena yield meagre or residual exceptional sets, underscoring the nuanced prevalence of BL in dynamics. The paper culminates with a new theorem showing a nonlinear two-step recurrence in which BL holds for almost all initial conditions but with a meagre exceptional set, highlighting both the ubiquity and subtlety of Benford behavior in nonlinear dynamical systems.

Abstract

This article provides a brief overview on a range of basic dynamical systems that conform to the logarithmic distribution of significant digits known as Benford's law. As presented here, most theorems are special cases of known, more general results about dynamical systems whose orbits or trajectories follow this logarithmic law, in one way or another. These results span a wide variety of systems: autonomous and non-autonomous; discrete- and continuous-time; one- and multi-dimensional; deterministic and stochastic. Illustrative examples include familiar systems such as the tent map, Newton's root-finding algorithm, and geometric Brownian motion. The treatise is informal, with the goal of showcasing to the specialists the generality and universal appeal of Benford's law throughout the mathematical field of dynamical systems. References to complete proofs are provided for each known result, while one new theorem is presented in some detail.

Figures (3)

• Figure A: Proportions (in percent) of first significant digits $D_1=1,2,\ldots, 9$ for the first $N= 10^4$ entries of $(x_n)$, for the first-order autonomous system $x_n = 2 x_{n-1}$ and $x_0=1$ (powers of $2$); the second-order linear system$x_n = x_{n-1} + x_{n-2}$ and $x_1=x_2=1$ (Fibonacci numbers); and the non-autonomous system $x_n = n x_{n-1}$ and $x_0=1$ (factorials). Each proportion matches quite closely the corresponding precise proportion in BL (bottom row), in the case of $(2^n)$ and $(F_n)$ quite strikingly so; see also Example \ref{['t14e1a']} below. This match is indicative of the fact that all three sequences indeed are Benford, according to Definition \ref{['def1']}.
• Figure B: Occurrences of $D_1$ for the Fibonacci numbers $F_1, F_2, \ldots , F_N$, for three different $N$. For comparison, the corresponding best approximation of the "Benford vector" $N \bigl(\log 2, \log \frac{3}{2} , \ldots , \log \frac{10}{9}\bigr)\in \mathbb{R}^9$ by $(N_1, N_2, \ldots, N_9)\in \mathbb{N}^9$ with $\sum_{j=1}^9 N_j=N$ is shown in italics.
• Figure C: For the "quadratic Fibonacci" recursion of Example \ref{['exPP1']}, that is, for $x_n = x_{n-1}^2 + x_{n-2}^2$, $n\ge 3$, the set $\mathbb{A}_0 = \{(x_1, x_2)\in \mathbb{R}^2: \lim_{n\to \infty}x_n=0 \}$ is non-empty, open, bounded and convex. Its boundary $\partial \mathbb{A}_0$ is a smooth oval containing the (unique) fixed point $(\frac{1}{2}, \frac{1}{2})$ of $T$. If $(x_1,x_2)\in \partial \mathbb{A}_0$ then $\lim_{n\to \infty}x_n = \frac{1}{2}$, and clearly $(x_n)$ is not Benford in this case. By contrast, Corollary \ref{['cor19b']} shows that $(x_n)$ is Benford for almost all, but not all, $(x_1,x_2)\in \mathbb{A}_0\cup \mathbb{A}_{\infty}$. The figure only displays slightly more than one quarter of $\mathbb{A}_0$ (light grey) and $\mathbb{A}_{\infty}$ (white), in light of their obvious symmetries; note, however, that neither set is symmetric w.r.t. the line $x_1=x_2$.

Theorems & Definitions (65)

• Definition 2.1
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Example 3.2
• Example 3.3
• Theorem 3.4
• proof
• Example 3.5