Benford's Law and Dynamical Systems

Dr. Arno Berger

Institute of Mechanics
Vienna University of Technology

April 9, 2003
202 Madison Hall
3:00 PM (Coffee and Cookies at 2:30)

Abstract

Significant digits in sufficiently large and diverse real-world aggregations of numerical data often are not equally likely but rather follow one particular logarithmic distribution. This observation, traditionally referred to as Benford's Law (BL), has been discussed extensively under a statistical as well as a mathematical perspective. Recent interest has focused on numerical data generated by dynamical systems, both in continuous and discrete time. Dynamical systems which obey BL are easily constructed. Contrary to ad-hoc constructions, however, it has been an open question whether large classes of dynamical systems may naturally show Benford's logarithmic distribution of significant digits. We present a fairly general, amazingly affirmative answer to this question for one-dimensional dynamical systems, and we discuss generalizations to higher dimensions. Our analysis proceeds through a careful combination of probabilistic and deterministic tehcniques, which in turn allows for a balanced appreciation of BL's astonishing ubiquity. We also explain why BL, at least in its strict form, should not be expected to hold for most of the classical "chaotic" systems, and we mention a number of challenging open questions.