Laws of Chaos

"Well written with ample examples and exercises... It fills an important gap in the literature." -Journal of the Indian Inst. of Science "The book provides a personal view on invariant measures and dynamical systems in one dimension. It is given a detailed study of the piecewise linear transformations under another spirit than that of {W. Doeblin} developed in the commemorative volume [Doeblin and modern probability. Proceedings of the Doeblin conference "50 years after Doeblin: Developments in the theory of Markov chains, Markov processes and sums of independent random variables", Contemporary Mathematics 149 (1993; articles are reviewed individually in Zbl )]. The book contains 13 chapters. Some titles are as follows: Spectral decomposition of the Frobenius-Perron operator, Markov transformations, Compactness theorem and Approximation of invariant densities, Stability of invariant measures, The inverse problem for the Frobenius-Perron equation and others. The style of the book is clear with good didactical perspectives for those who wish to study dynamical systems in connection with measure theory and ergodic theory. Finally, the book is a valuable contribution to the topic of Dynamical Systems." -Zentralblatt Math

• 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 Speedof 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 toSelected Problems.