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Smooth Ergodic Theory and Its Applications

Edited by: Anatole Katok Pennsylvania State University, University Park, PA
Rafael de la Llave University of Texas at Austin, Austin, TX
Yakov Pesin Pennsylvania State University, University Park, PA
Howard Weiss Pennsylvania State University, University Park, PA
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Hardcover ISBN: 978-0-8218-2682-9
Product Code: PSPUM/69
List Price: $190.00 MAA Member Price:$171.00
AMS Member Price: $152.00 Electronic ISBN: 978-0-8218-9374-6 Product Code: PSPUM/69.E List Price:$190.00
MAA Member Price: $171.00 AMS Member Price:$152.00
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List Price: $285.00 MAA Member Price:$256.50
AMS Member Price: $228.00 Click above image for expanded view Smooth Ergodic Theory and Its Applications Edited by: Anatole Katok Pennsylvania State University, University Park, PA Rafael de la Llave University of Texas at Austin, Austin, TX Yakov Pesin Pennsylvania State University, University Park, PA Howard Weiss Pennsylvania State University, University Park, PA Available Formats:  Hardcover ISBN: 978-0-8218-2682-9 Product Code: PSPUM/69  List Price:$190.00 MAA Member Price: $171.00 AMS Member Price:$152.00
 Electronic ISBN: 978-0-8218-9374-6 Product Code: PSPUM/69.E
 List Price: $190.00 MAA Member Price:$171.00 AMS Member Price: $152.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$285.00 MAA Member Price: $256.50 AMS Member Price:$228.00
• Book Details

Proceedings of Symposia in Pure Mathematics
Volume: 692001; 881 pp
MSC: Primary 11; 28; 34; 37; 53; 70;

During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student—or even an established mathematician who is not an expert in the area—to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools.

Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference.

Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincaré and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behavior in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behavior, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behavior of typical orbits, ergodicity, mixing, decay of correlations, and measure-theoretic entropy). Smooth ergodic theory also provides a foundation for numerous applications throughout mathematics (e.g., Riemannian geometry, number theory, Lie groups, and partial differential equations), as well as other sciences.

This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.

Graduate students and research mathematicians interested in ergodic theory and its applications.

• Part I. Lecture notes [ MR 1858533 ]
• L. Barreira and Ya. Pesin - Lectures on Lyapunov exponents and smooth ergodic theory [ MR 1858534 ]
• M. Brin - Appendix A: Hölder continuity of invariant distributions [ MR 1858534-a ]
• D. Dolgopyat, H. Hu and Ya Pesin - Appendix B: An example of a smooth hyperbolic measure with countably many ergodic components [ MR 1858534-b ]
• Anatole Katok - Cocycles, cohomology and combinatorial constructions in ergodic theory [ MR 1858535 ]
• Rafael de la Llave - A tutorial on KAM theory [ MR 1858536 ]
• Part II. Survey-expository articles [ MR 1858533 ]
• Part IIa. Systems with hyperbolic behavior [ MR 1858533 ]
• Viviane Baladi - Decay of correlations [ MR 1858537 ]
• Keith Burns, Charles Pugh, Michael Shub and Amie Wilkinson - Recent results about stable ergodicity [ MR 1858538 ]
• Huyi Hu - Statistical properties of some almost hyperbolic systems [ MR 1858539 ]
• Yuri Kifer - Random $f$-expansions [ MR 1858540 ]
• Mark Pollicott - Dynamical zeta functions [ MR 1858541 ]
• Jörg Schmeling and Howard Weiss - An overview of the dimension theory of dynamical systems [ MR 1858542 ]
• Grzegorz Świa̧tek - Collet-Eckmann condition in one-dimensional dynamics [ MR 1858543 ]
• Maciej P. Wojtkowski - Monotonicity, $\mathcal {J}$-algebra of Potapov and Lyapunov exponents [ MR 1858544 ]
• Part IIb. Geodesic flows [ MR 1858533 ]
• Patrick Eberlein - Geodesic flows in manifolds of nonpositive curvature [ MR 1858545 ]
• Gerhard Knieper - Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows [ MR 1858546 ]
• Part IIc. Algebraic systems and rigidity [ MR 1858533 ]
• Boris Kalinin and Anatole Katok - Invariant measures for actions of higher rank abelian groups [ MR 1858547 ]
• Dmitry Kleinbock - Some applications of homogeneous dynamics to number theory [ MR 1858548 ]
• Klaus Schmidt - Measurable rigidity of algebraic $\mathbb {Z}^d$-actions [ MR 1858549 ]
• Part IId. KAM-theory [ MR 1858533 ]
• L. H. Eliasson - Almost reducibility of linear quasi-periodic systems [ MR 1858550 ]
• Jürgen Pöschel - A lecture on the classical KAM theorem [ MR 1858551 ]
• M. Levi and J. Moser - A Lagrangian proof of the invariant curve theorem for twist mappings [ MR 1858552 ]
• Part III. Research articles [ MR 1858533 ]
• Jérôme Buzzi - Thermodynamical formalism for piecewise invertible maps: absolutely continuous invariant measures as equilibrium states [ MR 1858553 ]
• M. Guysinsky - Smoothness of holonomy maps derived from unstable foliation [ MR 1858554 ]
• Viorel Niţică and Frederico Xavier - Schrödinger operators and topological pressure on manifolds of negative curvature [ MR 1858555 ]
• Norbert Peyerimhoff - Isoperimetric and ergodic properties of horospheres in symmetric spaces [ MR 1858556 ]
• Alistair Windsor - Minimal but not uniquely ergodic diffeomorphisms [ MR 1858557 ]
• Michael Jakobson - Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions [ MR 1858558 ]
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 692001; 881 pp
MSC: Primary 11; 28; 34; 37; 53; 70;

During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student—or even an established mathematician who is not an expert in the area—to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools.

Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference.

Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincaré and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behavior in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behavior, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behavior of typical orbits, ergodicity, mixing, decay of correlations, and measure-theoretic entropy). Smooth ergodic theory also provides a foundation for numerous applications throughout mathematics (e.g., Riemannian geometry, number theory, Lie groups, and partial differential equations), as well as other sciences.

This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.

Graduate students and research mathematicians interested in ergodic theory and its applications.

• Part I. Lecture notes [ MR 1858533 ]
• L. Barreira and Ya. Pesin - Lectures on Lyapunov exponents and smooth ergodic theory [ MR 1858534 ]
• M. Brin - Appendix A: Hölder continuity of invariant distributions [ MR 1858534-a ]
• D. Dolgopyat, H. Hu and Ya Pesin - Appendix B: An example of a smooth hyperbolic measure with countably many ergodic components [ MR 1858534-b ]
• Anatole Katok - Cocycles, cohomology and combinatorial constructions in ergodic theory [ MR 1858535 ]
• Rafael de la Llave - A tutorial on KAM theory [ MR 1858536 ]
• Part II. Survey-expository articles [ MR 1858533 ]
• Part IIa. Systems with hyperbolic behavior [ MR 1858533 ]
• Viviane Baladi - Decay of correlations [ MR 1858537 ]
• Keith Burns, Charles Pugh, Michael Shub and Amie Wilkinson - Recent results about stable ergodicity [ MR 1858538 ]
• Huyi Hu - Statistical properties of some almost hyperbolic systems [ MR 1858539 ]
• Yuri Kifer - Random $f$-expansions [ MR 1858540 ]
• Mark Pollicott - Dynamical zeta functions [ MR 1858541 ]
• Jörg Schmeling and Howard Weiss - An overview of the dimension theory of dynamical systems [ MR 1858542 ]
• Grzegorz Świa̧tek - Collet-Eckmann condition in one-dimensional dynamics [ MR 1858543 ]
• Maciej P. Wojtkowski - Monotonicity, $\mathcal {J}$-algebra of Potapov and Lyapunov exponents [ MR 1858544 ]
• Part IIb. Geodesic flows [ MR 1858533 ]
• Patrick Eberlein - Geodesic flows in manifolds of nonpositive curvature [ MR 1858545 ]
• Gerhard Knieper - Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows [ MR 1858546 ]
• Part IIc. Algebraic systems and rigidity [ MR 1858533 ]
• Boris Kalinin and Anatole Katok - Invariant measures for actions of higher rank abelian groups [ MR 1858547 ]
• Dmitry Kleinbock - Some applications of homogeneous dynamics to number theory [ MR 1858548 ]
• Klaus Schmidt - Measurable rigidity of algebraic $\mathbb {Z}^d$-actions [ MR 1858549 ]
• Part IId. KAM-theory [ MR 1858533 ]
• L. H. Eliasson - Almost reducibility of linear quasi-periodic systems [ MR 1858550 ]
• Jürgen Pöschel - A lecture on the classical KAM theorem [ MR 1858551 ]
• M. Levi and J. Moser - A Lagrangian proof of the invariant curve theorem for twist mappings [ MR 1858552 ]
• Part III. Research articles [ MR 1858533 ]
• Jérôme Buzzi - Thermodynamical formalism for piecewise invertible maps: absolutely continuous invariant measures as equilibrium states [ MR 1858553 ]
• M. Guysinsky - Smoothness of holonomy maps derived from unstable foliation [ MR 1858554 ]
• Viorel Niţică and Frederico Xavier - Schrödinger operators and topological pressure on manifolds of negative curvature [ MR 1858555 ]
• Norbert Peyerimhoff - Isoperimetric and ergodic properties of horospheres in symmetric spaces [ MR 1858556 ]
• Alistair Windsor - Minimal but not uniquely ergodic diffeomorphisms [ MR 1858557 ]
• Michael Jakobson - Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions [ MR 1858558 ]
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.