Author | Boyarsky, Abraham. author |
---|---|
Title | Laws of Chaos [electronic resource] : Invariant Measures and Dynamical Systems in One Dimension / by Abraham Boyarsky, Paweล Gรณra |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1997 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2024-4 |
Descript | XVI, 400 p. online resource |
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 FrobeniusโPerron Operator -- 4.1 Motivation -- 4.2 Properties of the FrobeniusโPerron Operator -- 4.3 Representation of the FrobeniusโPerron Operator -- Problems for Chapter 4 -- 5. Absolutely Continuous Invariant Measures -- 5.1 Introduction -- 5.2 Existence of Absolutely Continuous Invariant Measures -- 5.3 LasotaโYorke Example of a Transformation with-out Absolutely Continuous Invariant Measure -- 5.4 Rychlikโs Theorem for Transformations with Countably Many Branches -- Problems for Chapter 5 -- 6. Other Existence Results -- 6.1 The Folklore Theorem -- 6.2 Rychlikโs Theorem for C1+? Transformations of the Interval -- 6.3 Piecewise Convex Transformations -- Problems for Chapter 6 -- 7. Spectral Decomposition of the FrobeniusโPerron Operator -- 7.1 Theorem of IonescuโTulcea and Marinescu -- 7.2 Quasi-Compactness of FrobeniusโPerron 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 FrobeniusโPerron 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 FrobeniusโPerron 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 FrobeniusโPerron Equation -- 12.1 The ErshovโMalinetskii 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 Kolodziejโs Proof of Ponceletโs Theorem -- Problems for Chapter 13 -- Solutions to Selected Problems