## Laws of Chaos: Invariant Measures and Dynamical Systems in One DimensionLaws of Chaos is an excellent pedagogical resource for studying probabilistic concepts in the analysis of one-dimensional chaotic systems. It combines three important areas of modern mathematics dynamical systems, measure theory, and ergodic theory, making it a truly up-to-date text for a graduate course for students of applied sciences. Chaotic systems arising from piecewise linear transformations are studied in detail since the probability density functions of such transformations can be easily found using linear algebra and provide a useful means to approximate the invariant measures of more complex dynamical systems. The book is rich in examples and graphical illustrations and includes over one hundred pages of problem sets and their solutions. It contains practical applications as diverse as random number generation, computer modeling, rotary drills, the pendulum, population dynamics, and projective geometry. It begins with brief reviews of the basics of measure theory and ergodic theory and is easily accessible to a student with one year of graduate studies and some exposure to linear algebra and measure theory. The combination of review, applications, and exercises with solutions also make it an ideal text for self study by a variety of applied scientists. Series: Probability and Its Applications Contents: Chapter I. Introduction 1.1 Overview 1.2 Examples of Piecewise Monotonic Transformations and Density Functions of the Absolutely Continuous Invariant Measures Chapter II. 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 Chapter III. 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 tau 3.6 Basic Constructions of Ergodic Theory 3.7 Infinite and Finite Invariant Measures Problems Chapter IV. The Frobenius-Perron Operator 4.1 Motivation 4.2 Properties of the Frobenius-Perron Operator 4.3 Representation of the Frobenius-Perron Operator Problems Chapter V. Absolutely Continuous Invariant Measures 5.1 Introduction 5.2 Existence of Absolutely Continuous Invariant Measures 5.3 Lasota-Yorke Example of a Transformation without an Absolutely Continuous Invariant Measure 5.4 Rychlik's Theorem for Transformations with Countably Many Branches Problems Chapter VI. Other Existence Results 6.1 The Folklore Theorem 6.2 Rychlik's Theorem for C 1+(varepsilon) Transformations of the Interval 6.3 Piecewise Convex Transformations Problems Chapter VII. 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 Chapter VIII. Properties of Absolutely Continuous Invariant Measures 8.1 Preliminary Results 8.2 Support of an Invariant Density 8.3 Speed Convergence of the Iterates of P n (tau)f 8.4 Bernoulli Property 8.5 Central Limit Theorem 8.6 Smoothness of the Density Function Problems Chapter IX. 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 Chapter X. 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 Chapter XI. 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 Chapter XII. The Inverse Problem for the Frobenius-Perron Equation 12.1 The Ershov-Malinetskii Result 12.2 Solving the Inverse Problem by Matrix Methods Chapter XIII. 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 Gipp Pendulum Regulator 13.5 Control of Chaotic Systems 13.6 Kolodziej's Proof of Poncelet's Theorem Problems Solutions to All Problems Bibliography Index |

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### Contents

Introduction | 1 |

Preliminaries | 7 |

Review of Ergodic Theory | 29 |

Copyright | |

14 other sections not shown

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Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension Abraham Boyarsky,Pawel Gora No preview available - 2012 |

### Common terms and phrases

absolutely continuous invariant absolutely continuous measure acim Birkhoff Ergodic Theorem Borel bounded variation Chapter compact condition construction continuous invariant measure converges countable Definition denote disjoint dynamical system eigenvalue eigenvector endpoints equal equation Ergodic Theorem Example fixed point follows Frobenius-Perron operator G BV G T(I graph Hence implies incidence matrix inequality integrable irreducible Lebesgue measure Lemma Let f Let us assume Let us define linear Markov transformation lower function Markov partition measurable set measure preserving measure space modulus nonsingular norm normalized measure space obtain orbits partition points piecewise expanding transformation piecewise linear Markov piecewise monotonic probability density function Problem properties Proposition prove quasi-compact r-invariant satisfies sequence shown in Figure Solution space and let spectral radius stochastic kernel subintervals subset summand tion topologically conjugate trajectory uniformly bounded weak topology weakly mixing