This monograph offers a coherent, self-contained account of the theory of Sinai–Ruelle–Bowen measures and decay of correlations for nonuniformly hyperbolic dynamical systems. A central topic in the statistical theory of dynamical systems, the book in particular provides a detailed exposition of the theory developed by L.-S. Young for systems admitting induced maps with certain analytic and geometric properties. After a brief introduction and preliminary results, Chapters 3, 4, 6 and 7 provide essentially the same pattern of results in increasingly interesting and complicated settings. Each chapter builds on the previous one, apart from Chapter 5 which presents a general abstract framework to bridge the more classical expanding and hyperbolic systems explored in Chapters 3 and 4 with the nonuniformly expanding and partially hyperbolic systems described in Chapters 6 and 7. Throughout the book, the theory is illustrated with applications. A clear and detailed account of topics of current research interest, this monograph will be of interest to researchers in dynamical systems and ergodic theory. In particular, beginning researchers and graduate students will appreciate the accessible, self-contained presentation.

A self-contained, comprehensive account of modern smooth ergodic theory, the mathematical foundation of deterministic chaos.

This EMS volume, the first edition of which was published as Dynamical Systems II, EMS 2, familiarizes the reader with the fundamental ideas and results of modern ergodic theory and its applications to dynamical systems and statistical mechanics. The enlarged and revised second edition adds two new contributions on ergodic theory of flows on homogeneous manifolds and on methods of algebraic geometry in the theory of interval exchange transformations.

This book gives a comprehensive overview of the relationship between admissibility and hyperbolicity. Essential theories and selected developments are discussed with highlights to applications. The dedicated readership includes researchers and graduate students specializing in differential equations and dynamical systems (with emphasis on hyperbolicity) who wish to have a broad view of the topic and working knowledge of its techniques. The book may also be used as a basis for appropriate graduate courses on hyperbolicity; the pointers and references given to further research will be particularly useful. The material is divided into three parts: the core of the theory, recent developments, and applications. The first part pragmatically covers the relation between admissibility and hyperbolicity, starting with the simpler case of exponential contractions. It also considers exponential dichotomies, both for discrete and continuous time, and establishes corresponding results building on the arguments for exponential contractions. The second part considers various extensions of the former results, including a general approach to the construction of admissible spaces and the study of nonuniform exponential behavior. Applications of the theory to the robustness of an exponential dichotomy, the characterization of hyperbolic sets in terms of admissibility, the relation between shadowing and structural stability, and the characterization of hyperbolicity in terms of Lyapunov sequences are given in the final part.

This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey "Principal Structures of Volume 1A.The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations).. Written by experts in the field.. The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.

The main objective of this book is to give a broad uni?ed introduction to the study of dimension and recurrence inhyperbolic dynamics. It includes a disc- sion of the foundations, main results, and main techniques in the rich interplay of fourmain areas of research: hyperbolic dynamics, dimension theory, multifractal analysis, and quantitative recurrence. It also gives a panorama of several selected topics of current research interest. This includes topics on irregular sets, var- tional principles, applications to number theory, measures of maximal dimension, multifractal rigidity, and quantitative recurrence. The book isdirected to researchersas well as graduate students whowish to have a global view of the theory together with a working knowledgeof its main techniques. It can also be used as a basis for graduatecourses in dimension theory of dynamical systems, multifractal analysis (together with a discussion of several special topics), and pointwise dimension and recurrence in hyperbolic dynamics. I hope that the book may serve as a fast entry point to this exciting and active ?eld of research, and also that it may lead to further developments.

Mixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification? The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book. The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.