Laws of Chaos

A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the founda­ tions of Newtonian physics and opened a window to the modern theory of nonlinear dynamics and chaos. Although in the 1930s and 1940s strange behavior was observed in many physical systems, the notion that this phenomenon was inherent in deterministic systems was never suggested. Even with the powerful results of S. Smale in the 1960s, complicated be­ havior of deterministic systems remained no more than a mathematical curiosity. Not until the late 1970s, with the advent of fast and cheap comput­ ers, was it recognized that chaotic behavior was prevalent in almost all domains of science and technology. Smale horseshoes began appearing in many scientific fields. In 1971, the phrase 'strange attractor' was coined to describe complicated long-term behavior of deterministic systems, and the term quickly became a paradigm of nonlinear dynamics. The tools needed to study chaotic phenomena are entirely different from those used to study periodic or quasi-periodic systems; these tools are analytic and measure-theoretic rather than geometric. For example, in throwing a die, we can study the limiting behavior of the system by viewing the long-term behavior of individual orbits. This would reveal incomprehensibly complex behavior. Or we can shift our perspective: Instead of viewing the long-term outcomes themselves, we can view the probabilities of these outcomes. This is the measure-theoretic approach taken in this book.

Contenu

1. Introduction.- 1.1 Overview.- 1.2 Examples of Piecewise Monotonic Transformations and the Density Functions of Absolutely Continuous Invariant Measures.- 2. Preliminaries.- 2.1 Review of Measure Theory.- 2.2 Spaces of Functions and Measures.- 2.3 Functions of Bounded Variation in One Dimension.- 2.4 Conditional Expectations.- Problems for Chapter 2.- 3. Review of Ergodic Theory.- 3.1 Measure-Preserving Transformations.- 3.2 Recurrence and Ergodicity.- 3.3 The Birkhoff Ergodic Theorem.- 3.4 Mixing and Exactness.- 3.5 The Spectrum of the Koopman Operator and the Ergodic Properties of ?.- 3.6 Basic Constructions of Ergodic Theory.- 3.7 Infinite and Finite Invariant Measures.- Problems for Chapter 3.- 4. The Frobenius-Perron Operator.- 4.1 Motivation.- 4.2 Properties of the Frobenius-Perron Operator.- 4.3 Representation of the Frobenius-Perron Operator.- Problems for Chapter 4.- 5. Absolutely Continuous Invariant Measures.- 5.1 Introduction.- 5.2 Existence of Absolutely Continuous Invariant Measures.- 5.3 Lasota-Yorke Example of a Transformation with-out Absolutely Continuous Invariant Measure.- 5.4 Rychlik's Theorem for Transformations with Countably Many Branches.- Problems for Chapter 5.- 6. Other Existence Results.- 6.1 The Folklore Theorem.- 6.2 Rychlik's Theorem for C1+? Transformations of the Interval.- 6.3 Piecewise Convex Transformations.- Problems for Chapter 6.- 7. Spectral Decomposition of the Frobenius-Perron Operator.- 7.1 Theorem of Ionescu-Tulcea and Marinescu.- 7.2 Quasi-Compactness of Frobenius-Perron Operator.- 7.3 Another Approach to Spectral Decomposition: Constrictiveness.- Problems for Chapter 7.- 8. Properties of Absolutely Continuous Invariant Measures.- 8.1 Preliminary Results.- 8.2 Support of an Invariant Density.- 8.3 Speed of Convergence of the Iterates of Pn?f.- 8.4 Bernoulli Property.- 8.5 Central Limit Theorem.- 8.6 Smoothness of the Density Function.- Problems for Chapter 8.- 9. Markov Transformations.- 9.1 Definitions and Notation.- 9.2 Piecewise Linear Markov Transformations and the Matrix Representation of the Frobenius-Perron Operator.- 9.3 Eigenfunctions of Matrices Induced by Piecewise Linear Markov Transformations.- 9.4 Invariant Densities of Piecewise Linear Markov Transformations.- 9.5 Irreducibility and Primitivity of Matrix Representations of Frobenius-Perron Operators.- 9.6 Bounds on the Number of Ergodic Absolutely Continuous Invariant Measures.- 9.7 Absolutely Continuous Invariant Measures that Are Maximal.- Problems for Chapter 9.- 10. Compactness Theorem and Approximation of Invariant Densities.- 10.1 Introduction.- 10.2 Strong Compactness of Invariant Densities.- 10.3 Approximation by Markov Transformations.- 10.4 Application to Matrices: Compactness of Eigenvectors for Certain Non-Negative Matrices.- 11. Stability of Invariant Measures.- 11.1 Stability of a Linear Stochastic Operator.- 11.2 Deterministic Perturbations of Piecewise Expanding Transformations.- 11.3 Stochastic Perturbations of Piecewise Expanding Transformations.- Problems for Chapter 11.- 12. The Inverse Problem for the Frobenius-Perron Equation.- 12.1 The Ershov-Malinetskii Result.- 12.2 Solving the Inverse Problem by Matrix Methods.- 13. Applications.- 13.1 Application to Random Number Generators.- 13.2 Why Computers Like Absolutely Continuous Invariant Measures.- 13.3 A Model for the Dynamics of a Rotary Drill.- 13.4 A Dynamic Model for the Hipp Pendulum Regulator.- 13.5 Control of Chaotic Systems.- 13.6 Kolodziej's Proof of Poncelet's Theorem.- Problems for Chapter 13.- Solutions to Selected Problems.