Proceedings of Symposia in Pure Mathematics 2001; 881 pp; hardcover Volume: 69 ISBN10: 0821826824 ISBN13: 9780821826829 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 studentor even an established mathematician who is not an expert in the areato 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. Readership Graduate students and research mathematicians interested in ergodic theory and its applications. Table of Contents 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
Surveyexpository 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  ColletEckmann condition in onedimensional 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
KAMtheory  L. H. Eliasson  Almost reducibility of linear quasiperiodic 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
