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Book DetailsProceedings of Symposia in Pure MathematicsVolume: 69; 2001; 881 ppMSC: 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 minicourses 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 nonlinear 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 measuretheoretic 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 twofold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a stateoftheart 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 selfcontained introduction to a particular area of smooth ergodic theory; thematic sections based on minicourses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.ReadershipGraduate students and research mathematicians interested in ergodic theory and its applications.

Table of Contents

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 1858534a ]

D. Dolgopyat, H. Hu and Ya Pesin  Appendix B: An example of a smooth hyperbolic measure with countably many ergodic components [ MR 1858534b ]

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. Surveyexpository 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  ColletEckmann condition in onedimensional 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. KAMtheory [ MR 1858533 ]

L. H. Eliasson  Almost reducibility of linear quasiperiodic 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 ]


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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 minicourses 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 nonlinear 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 measuretheoretic 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 twofold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a stateoftheart 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 selfcontained introduction to a particular area of smooth ergodic theory; thematic sections based on minicourses 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 1858534a ]

D. Dolgopyat, H. Hu and Ya Pesin  Appendix B: An example of a smooth hyperbolic measure with countably many ergodic components [ MR 1858534b ]

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. Surveyexpository 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  ColletEckmann condition in onedimensional 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. KAMtheory [ MR 1858533 ]

L. H. Eliasson  Almost reducibility of linear quasiperiodic 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 ]