A brief survey of Benford's Law in dynamical systems

TL;DR

This paper surveys how Benford's law applies across various dynamical systems, highlighting its universality and providing examples from different types of systems, with one new theorem introduced.

Contribution

It offers a comprehensive overview of Benford's law in dynamical systems, including a new theorem and numerous known results with proofs.

Findings

Benford's law applies to a wide variety of dynamical systems.

Most theorems are special cases of more general results.

The paper presents one new theorem in detail.

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.