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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, TX, and Yakov Pesin and Howard Weiss, Pennsylvania State University, University Park, PA
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Proceedings of Symposia in Pure Mathematics
2001; 881 pp; hardcover
Volume: 69
ISBN-10: 0-8218-2682-4
ISBN-13: 978-0-8218-2682-9
List Price: US$190 Member Price: US$152
Order Code: PSPUM/69

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.

Lecture notes
• L. Barreira and Ya. Pesin -- Lectures on Lyapunov exponents and smooth ergodic theory
• A. Katok and E. A. Robinson, Jr. -- Cocycles, cohomology and combinatorial constructions in ergodic theory
• R. de la Llave -- A tutorial on KAM theory
Survey-expository articles
• A. Katok -- Systems with hyperbolic behavior
• V. Baladi -- Decay of correlations
• K. Burns, C. Pugh, M. Shub, and A. Wilkinson -- Recent results about stable ergodicity
• H. Hu -- Statistical properties of some almost hyperbolic systems
• Y. Kifer -- Random $$f$$-expansions
• M. Pollicott -- Dynamical zeta functions
• J. Schmeling and H. Weiss -- An overview of the dimension theory of dynamical systems
• G. Światek -- Collet-Eckmann condition in one-dimensional dynamics
• M. P. Wojtkowski -- Monotonicity, $$\mathcal J$$-algebra of Potapov and Lyapunov exponents
Geodesic flows
• P. Eberlein -- Geodesic flows in manifolds of nonpositive curvature
• G. Knieper -- Closed geodesics and the uniqueness of the maximal measure for rank 1 geodesic flows
Algebraic systems and rigidity
• B. Kalinin and A. Katok -- Invariant measures for actions of higher rank abelian groups
• D. Kleinbock -- Some applications of homogeneous dynamics to number theory
• K. Schmidt -- Measurable rigidity of algebraic $$\mathbb {Z}^d$$-actions
KAM-theory
• L. H. Eliasson -- Almost reducibility of linear quasi-periodic systems
• J. Pöschel -- A lecture on the classical KAM theorem
• M. Levi and J. Moser -- A Lagrangian proof of the invariant curve theorem for twist mappings
Research articles
• J. Buzzi -- Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states
• M. Guysinsky -- Smoothness of holonomy maps derived from unstable foliation
• V. Niţică and F. Xavier -- Schrödinger operators and topological pressure on manifolds of negative curvature
• N. Peyerimhoff -- Isoperimetric and ergodic properties of horospheres in symmetric spaces
• A. Windsor -- Minimal but not uniquely ergodic diffeomorphisms
• M. Jakobson -- Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions