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Hardcover ISBN:  9781470470654 
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Book DetailsGraduate Studies in MathematicsVolume: 231; 2023; 336 ppMSC: Primary 37
This is a Revised Edition of: GSM/148
This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided.
In this second edition, the authors improved the exposition and added more exercises to make the book even more studentoriented. They also added new material to bring the book more in line with the current research in dynamical systems.
ReadershipGraduate students and researchers interested in smooth ergodic theory and ordinary differential equations.

Table of Contents

The core of the theory

Examples of hyperbolic dynamical systems

General theory of Lyapunov exponents

Cocylces over dynamical systems

The multiplicative ergodic theorem

Elements of the nonuniform hyperbolicity theory

Lyapunov stability theory of nonautonomous equations

Local manifold theory

Absolute continuity of local manifolds

Ergodic properties of smooth hyperbolic measures

Geodesic flows on surfaces of nonpositive curvature

Topological and ergodic properties of hyperbolic measures

Selected advanced topics

Cone techniques

Partially hyperbolic diffeomorphisms with nonzero exponents

More examples of dynamical systems with nonzero Lyapunov exponents

Anosov rigidity

$C^1$ pathological behavior: Pugh’s example


Additional Material

Reviews

The authors — both of them major contributors to the field — note that the theory has many applications inside and outside of mathematics (from probability and statistics to physics, biology, and beyond). Their goal in this new edition is to expand and improve the exposition with more details and informal discussions. All major results are proved. They have expanded the scope to bring students closer to modern research in dynamics.
Bill Satzer, MAA Reviews


RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
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 Additional Material
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This is a Revised Edition of: GSM/148
This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided.
In this second edition, the authors improved the exposition and added more exercises to make the book even more studentoriented. They also added new material to bring the book more in line with the current research in dynamical systems.
Graduate students and researchers interested in smooth ergodic theory and ordinary differential equations.

The core of the theory

Examples of hyperbolic dynamical systems

General theory of Lyapunov exponents

Cocylces over dynamical systems

The multiplicative ergodic theorem

Elements of the nonuniform hyperbolicity theory

Lyapunov stability theory of nonautonomous equations

Local manifold theory

Absolute continuity of local manifolds

Ergodic properties of smooth hyperbolic measures

Geodesic flows on surfaces of nonpositive curvature

Topological and ergodic properties of hyperbolic measures

Selected advanced topics

Cone techniques

Partially hyperbolic diffeomorphisms with nonzero exponents

More examples of dynamical systems with nonzero Lyapunov exponents

Anosov rigidity

$C^1$ pathological behavior: Pugh’s example

The authors — both of them major contributors to the field — note that the theory has many applications inside and outside of mathematics (from probability and statistics to physics, biology, and beyond). Their goal in this new edition is to expand and improve the exposition with more details and informal discussions. All major results are proved. They have expanded the scope to bring students closer to modern research in dynamics.
Bill Satzer, MAA Reviews