A brief survey of Benford's Law in 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.