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The Resource Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension, edited by Abraham Boyarsky, Pawel Gora, (electronic resource)
Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension, edited by Abraham Boyarsky, Pawel Gora, (electronic resource)
Resource Information
The item Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension, edited by Abraham Boyarsky, Pawel Gora, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.This item is available to borrow from all library branches.
Resource Information
The item Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension, edited by Abraham Boyarsky, Pawel Gora, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.
This item is available to borrow from all library branches.
 Summary
 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 longterm 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 quasiperiodic systems; these tools are analytic and measuretheoretic rather than geometric. For example, in throwing a die, we can study the limiting behavior of the system by viewing the longterm behavior of individual orbits. This would reveal incomprehensibly complex behavior. Or we can shift our perspective: Instead of viewing the longterm outcomes themselves, we can view the probabilities of these outcomes. This is the measuretheoretic approach taken in this book
 Language

 eng
 eng
 Edition
 1st ed. 1997.
 Extent
 1 online resource (XVI, 400 p.)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Contents

 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 MeasurePreserving 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 without 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 QuasiCompactness 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 NonNegative 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
 Isbn
 9781461220244
 Label
 Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension
 Title
 Laws of Chaos
 Title remainder
 Invariant Measures and Dynamical Systems in One Dimension
 Statement of responsibility
 edited by Abraham Boyarsky, Pawel Gora
 Language

 eng
 eng
 Summary
 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 longterm 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 quasiperiodic systems; these tools are analytic and measuretheoretic rather than geometric. For example, in throwing a die, we can study the limiting behavior of the system by viewing the longterm behavior of individual orbits. This would reveal incomprehensibly complex behavior. Or we can shift our perspective: Instead of viewing the longterm outcomes themselves, we can view the probabilities of these outcomes. This is the measuretheoretic approach taken in this book
 Dewey number
 519.2
 http://bibfra.me/vocab/relation/httpidlocgovvocabularyrelatorsedt

 KgnhKhG0RM
 HS0DPcWFIVI
 Image bit depth
 0
 Language note
 English
 LC call number

 QA273.A1274.9
 QA274274.9
 Literary form
 non fiction
 Nature of contents
 dictionaries
 http://library.link/vocab/relatedWorkOrContributorName

 Boyarsky, Abraham.
 Gora, Pawel.
 Series statement
 Probability and Its Applications,
 http://library.link/vocab/subjectName

 Distribution (Probability theory
 Differentiable dynamical systems
 Mathematics
 Probability Theory and Stochastic Processes
 Dynamical Systems and Ergodic Theory
 Applications of Mathematics
 Label
 Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension, edited by Abraham Boyarsky, Pawel Gora, (electronic resource)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Antecedent source
 mixed
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Color
 not applicable
 Content category
 text
 Content type code

 txt
 Contents
 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 MeasurePreserving 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 without 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 QuasiCompactness 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 NonNegative 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
 Dimensions
 unknown
 Edition
 1st ed. 1997.
 Extent
 1 online resource (XVI, 400 p.)
 File format
 multiple file formats
 Form of item
 online
 Isbn
 9781461220244
 Level of compression
 uncompressed
 Media category
 computer
 Media type code

 c
 Other control number
 10.1007/9781461220244
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number

 (CKB)3400000000089805
 (SSID)ssj0001297352
 (PQKBManifestationID)11847562
 (PQKBTitleCode)TC0001297352
 (PQKBWorkID)11362433
 (PQKB)10992091
 (DEHe213)9781461220244
 (MiAaPQ)EBC3075697
 (EXLCZ)993400000000089805
 Label
 Laws of Chaos : Invariant Measures and Dynamical Systems in One Dimension, edited by Abraham Boyarsky, Pawel Gora, (electronic resource)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Antecedent source
 mixed
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Color
 not applicable
 Content category
 text
 Content type code

 txt
 Contents
 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 MeasurePreserving 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 without 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 QuasiCompactness 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 NonNegative 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
 Dimensions
 unknown
 Edition
 1st ed. 1997.
 Extent
 1 online resource (XVI, 400 p.)
 File format
 multiple file formats
 Form of item
 online
 Isbn
 9781461220244
 Level of compression
 uncompressed
 Media category
 computer
 Media type code

 c
 Other control number
 10.1007/9781461220244
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number

 (CKB)3400000000089805
 (SSID)ssj0001297352
 (PQKBManifestationID)11847562
 (PQKBTitleCode)TC0001297352
 (PQKBWorkID)11362433
 (PQKB)10992091
 (DEHe213)9781461220244
 (MiAaPQ)EBC3075697
 (EXLCZ)993400000000089805
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