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Author: Giovanni Forni Publisher: American Mathematical Soc. ISBN: 9780821871546 Category : Mathematics Languages : en Pages : 356
Book Description
This volume collects a set of contributions by participants of the Workshop Partially hyperbolic dynamics, laminations, and Teichmuller flow held at the Fields Institute in Toronto in January 2006. The Workshop brought together several leading experts in two very active fields of contemporary dynamical systems theory: partially hyperbolic dynamics and Teichmuller dynamics. They are unified by ideas coming from the theory of laminations and foliations, dynamical hyperbolicity, and ergodic theory. These are the main themes of the current volume. The volume contains both surveys and research papers on non-uniform and partial hyperbolicity, on dominated splitting and beyond (in Part I), Teichmuller dynamics with applications to interval exchange transformations and on the topology of moduli spaces of quadratic differentials (in Part II), foliations and laminations and other miscellaneous papers (in Part III). Taken together these papers provide a snapshot of the state of the art in some of the most active topics at the crossroads between dynamical systems, smooth ergodic theory, geometry and topology, suitable for advanced graduate students and researchers.Non-specialists will find the extensive, in-depth surveys especially useful.
Author: Giovanni Forni Publisher: American Mathematical Soc. ISBN: 9780821871546 Category : Mathematics Languages : en Pages : 356
Book Description
This volume collects a set of contributions by participants of the Workshop Partially hyperbolic dynamics, laminations, and Teichmuller flow held at the Fields Institute in Toronto in January 2006. The Workshop brought together several leading experts in two very active fields of contemporary dynamical systems theory: partially hyperbolic dynamics and Teichmuller dynamics. They are unified by ideas coming from the theory of laminations and foliations, dynamical hyperbolicity, and ergodic theory. These are the main themes of the current volume. The volume contains both surveys and research papers on non-uniform and partial hyperbolicity, on dominated splitting and beyond (in Part I), Teichmuller dynamics with applications to interval exchange transformations and on the topology of moduli spaces of quadratic differentials (in Part II), foliations and laminations and other miscellaneous papers (in Part III). Taken together these papers provide a snapshot of the state of the art in some of the most active topics at the crossroads between dynamical systems, smooth ergodic theory, geometry and topology, suitable for advanced graduate students and researchers.Non-specialists will find the extensive, in-depth surveys especially useful.
Author: Luís Barreira Publisher: American Mathematical Society ISBN: 1470473070 Category : Mathematics Languages : en Pages : 355
Book Description
This book is the first comprehensive introduction to smooth ergodic theory. It consists of two parts: the first introduces the core of the theory and the second discusses more advanced topics. In particular, the book describes the general theory of Lyapunov exponents and its applications to the stability theory of differential equations, the concept of nonuniform hyperbolicity, stable manifold theory (with emphasis on absolute continuity of invariant foliations), and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. A detailed description of all the basic examples of conservative systems with nonzero Lyapunov exponents, including the geodesic flows on compact surfaces of nonpositive curvature, is also presented. There are more than 80 exercises. The book is aimed at graduate students specializing in dynamical systems and ergodic theory as well as anyone who wishes to get a working knowledge of smooth ergodic theory and to learn how to use its tools. It can also be used as a source for special topics courses on nonuniform hyperbolicity. The only prerequisite for using this book is a basic knowledge of real analysis, measure theory, differential equations, and topology, although the necessary background definitions and results are provided. In this second edition, the authors improved the exposition and added more exercises to make the book even more student-oriented. They also added new material to bring the book more in line with the current research in dynamical systems.
Author: Robert Ellis Publisher: American Mathematical Soc. ISBN: 0821806084 Category : Mathematics Languages : en Pages : 348
Book Description
This book is a very readable exposition of the modern theory of topological dynamics and presents diverse applications to such areas as ergodic theory, combinatorial number theory and differential equations. There are three parts: 1) The abstract theory of topological dynamics is discussed, including a comprehensive survey by Furstenberg and Glasner on the work and influence of R. Ellis. Presented in book form for the first time are new topics in the theory of dynamical systems, such as weak almost-periodicity, hidden eigenvalues, a natural family of factors and topological analogues of ergodic decomposition. 2) The power of abstract techniques is demonstrated by giving a very wide range of applications to areas of ergodic theory, combinatorial number theory, random walks on groups and others. 3) Applications to non-autonomous linear differential equations are shown. Exposition on recent results about Floquet theory, bifurcation theory and Lyapanov exponents is given.
Author: Benson Farb Publisher: American Mathematical Soc. ISBN: 0821838385 Category : Mathematics Languages : en Pages : 384
Book Description
The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.
Author: Benson Farb Publisher: American Mathematical Soc. ISBN: 0821898876 Category : Mathematics Languages : en Pages : 371
Book Description
Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Author: Boris Hasselblatt Publisher: Springer ISBN: 3319430599 Category : Mathematics Languages : en Pages : 334
Book Description
Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation.
Author: A. B. Katok Publisher: American Mathematical Soc. ISBN: 0821826824 Category : Mathematics Languages : en Pages : 895
Book Description
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 Poincare 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 corre 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.
Author: Luis Barreira Publisher: American Mathematical Soc. ISBN: 0821829211 Category : Mathematics Languages : en Pages : 166
Book Description
A systematic introduction to the core of smooth ergodic theory. An expanded version of an earlier work by the same authors, it describes the general (abstract) theory of Lyapunov exponents and the theory's applications to the stability theory of differential equations, the stable manifold theory, absolute continuity of stable manifolds, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows). It could be used as a primary text for a course on nonuniform hyperbolic theory or as supplemental reading for a course on dynamical systems. Assumes a basic knowledge of real analysis, measure theory, differential equations, and topology. c. Book News Inc.
Author: Anatole Katok Publisher: Cambridge University Press ISBN: 9780521575577 Category : Mathematics Languages : en Pages : 828
Book Description
This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.
Author: Giovanni Forni
Author: Giovanni Forni
Author: Luís Barreira
Author: Robert Ellis
Author: Benson Farb
Author: Benson Farb
Author: Boris Hasselblatt
Author: A. B. Katok
Author: Luis Barreira
Author: Anatole Katok
Author: Luis Barreira