"In recent years there has been an explosive growth in the study of physical, biological, and economic systems that can be profitably studies using densities. Because of the general inaccessibility of the mathematical literature to the nonspecialist, little diffusion of the applicable mathematics into the study of these ""chaotic"" systems has taken place. This book will help bridge that gap. To show how densities arise in simple deterministic systems, the authors give a unified treatment of a variety of mathematical system generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from many fields to illustrate the utility of the concepts and techniques presented, and the ideas in this book should thus prove useful in the study of a number of applied sciences. The authors assume that the reader has a knowledge of advanced calculus and differential equations. Basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed. Physicists, chemists, and biomathematicians studying chaotic behavior will find this book of value. It will also be a useful reference or text for mathematicians and graduate students working in ergodic theory and dynamical systems."

The revised edition of Probabilistic Properties of Deterministic Systems (Cambridge U. Press, 1985). A text-reference for physicists, chemists, and biomathematicians, which presents a unified treatment of a variety of mathematical systems that generate densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples are drawn from a variety of the sciences to illustrate the utility of the techniques presented. Assumes a knowledge of advanced calculus and differential equations.
Annotation c. Book News, Inc., Portland, OR (booknews.com)

"In the second edition of their book, the authors have added one section on discrete time systems with multiplicative perturbations and six sections dealing with the property of sweeping and the corresponding Foguel alternative in various contexts (Markov operators, stochastic semigroups in continuous time, solutions of the Fokker-Planck equation). A completely new chapter includes sections on Markov and Foias operators, several notions of asymptotic stability for dynamical systems as well as a final section devoted to iterated function systems and fractals. In addition, the book now contains almost 100 exercises. ZENTRALBLATT MATH"

This book gives a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from a variety of the sciences to illustrate the utility of the techniques presented. This material was organized and written to be accessible to scientists with knowledge of advanced calculus and differential equations. In various concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and chastic integrals and differential equations are introduced. 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 intends to bridge that gap.

Preface to the 2nd edition
x Preface to the 1st edition
x 1 Introduction
x 2 The Toolbox
x 3 Markov and Frobenius-Perron Operators
x 4 Studying Chaos with Densities
x 5 The Asymptotic Properties of Densities
x 6 The Behavior of Transformations on Intervals and Manifolds
x 7 Continuous Time Systems: An Introduction
x 8 Discrete Time Processes Embedded in Continuous Time Systems
x 9 Entropy
x 10 Stochastic Perturbation of Discrete Time Systems
x 11 Stochastic Perturbation of Continuous Time Systems
x 12 Markov and Foias Operators
x References
x Notation and Symbols
x Index