This book is about densities. In the history of science, the concept of densities
emerged only recently as attempts were made to provide unifying descriptions of
phenomena that appeared to be statistical in nature. Thus, for example, the
introduction of the Maxwellian velocity distribution rapidly led to a unification
of dilute gas theory; quantum mechanics developed from attempts to justify
Planck's ad hoc derivation of the equation for the density of blackbody radiation;
and the field of human demography grew rapidly after the introduction of the
Gompertzian age distribution.
From these and many other examples, as well as the formal development of
probability and statistics, we have come to associate the appearance of densities
with the description of large systems containing inherent elements of uncertainty.
Viewed from this perspective one might find it surprising to pose the question:
"What is the smallest number of elements that a system must have, and how much
uncertainty must exist, before a description in terms of densities becomes useful
and/or necessary?" The answer is surprising, and runs counter to the intuition of
many. A one-dimensional system containing only one object whose dynamics are
completely deterministic (no uncertainty) can generate a density of states! This
fact has only become apparent in the past half-century due to the pioneering work
of E. Borel [1909], A. Renyi [1957], and S. Ulam and J. von Neumann. These
results, however, are not generally known outside that small group of mathe-
mathematicians working in ergodic theory.
The past few years have witnessed an explosive growth in interest in physical,
biological, and economic systems that could be profitably studied using densities.
Due to the general inaccessibility of the mathematical literature to the non-
mathematician, there has been little diffusion of the concepts and techniques from
ergodic theory into the study of these "chaotic" systems. This book attempts to
bridge that gap.