Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents

Description

Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Among other things, this provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos - the appearance of 'chaotic' motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichmuller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schroedinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

Luis Barreira is a Professor of Mathematics at Instituto Superior Tecnico in Lisbon. He obtained his PhD from The Pennsylvania State University in 1996, under the guidance of Yakov Pesin, with whom he co-authored the book Lyapunov Exponents and Smooth Ergodic Theory. He has also written two surveys and more than 40 research papers in dynamical systems. Yakov Pesin is a Distinguished Professor of Mathematics at The Pennsylvania State University. He obtained his PhD from The Gorky State University in 1979. He is the author of three books, Dimension Theory in Dynamical Systems, Lectures on Partial Hyperbolicity and Stable Ergodicity, and, with Luis Barreira, Lyapunov Exponents and Smooth Ergodic Theory, as well as six surveys and more than 70 research papers. He is an executive editor of the journal Ergodic Theory and Dynamical Systems.

'... will be indispensable for any mathematically inclined reader with a serious interest in the subject.' EMS Newsletter