with a supplement by Anatole Katok and Leonardo Mendoza

ISBN 0-521-34187-6, Paperback, 1997: ISBN 0-521-57557-5.

The table of
contents
and
preface
can be read here. Some
corrigenda
are available, including a major correction on
page 254ff. Please
report
any errors you notice in the book to ```
```

.
Note one serious omission: The first three print runs
(up to the first paperback printing) fail to acknowledge that Section
20.6. reproduces work of **Charles Hansen Toll** (A multiplicative asymptotic of
the prime geodesic theorem, Thesis, University of Maryland 1984). Our
sincere apologies for this failure to give due credit.

The theory of dynamical systems is a major mathematical discipline closely intertwined with all main areas of mathematics. Its concepts, methods and paradigms greatly stimulate research in many sciences and gave rise to the vast new area variously called applied dynamics, nonlinear science, or chaos theory. This book provides the first self-contained coherent comprehensive exposition of the theory of dynamical systems as a core mathematical discipline while providing researchers interested in applications with fundamental tools and paradigms. It introduces and rigorously develops the central concepts and methods in dynamical systems in a hands-on fashion.

It starts with a comprehensive discussion of a series of elementary but fundamental examples. These are used to formulate the general program of the study of asymptotic properties as well as to introduce the principal notions (differentiable and topological equivalence, moduli, asymptotic orbit growth, entropies, ergodicity, etc.) and, in a simplified way, a number of important methods (fixed point methods, coding, KAM-type Newton method, local normal forms, etc.).

The main theme of the second part is the interplay between local analysis near individual (e.g., periodic) orbits and the global complexity of the orbit structure. This is achieved by exploring hyperbolicity, transversality, global topological invariants, and variational methods. The methods include study of stable and unstable manifolds, bifurcations, index and degree, and construction of orbits as minima and minimaxes of action functionals.

In the third and fourth part the general program is carried out for low-dimensional and hyperbolic dynamical systems which are particularly amenable to such analysis. In addition these systems have interesting particular properties. For hyperbolic systems there are structural stability, theory of equilibrium (Gibbs) measures, and asymptotic distribution of periodic orbits, in low-dimensional dynamical systems classical Poincaré-Denjoy theory, and Poincaré-Bendixson theories are presented as well as more recent developments, including the theory of twist maps, interval exchange transformations and noninvertible interval maps.

This book provides a large number of systematic exercises in order to be the principal source for the professional training of future researchers. On the other hand the book may be used by advanced undergraduates in mathematics, graduate students in any area of the mathematical sciences and graduate students in science and engineering with a strong mathematical background as well as researchers in any area of mathematics, science or engineering. Since a considerable part of the material of the book is either previously unpublished or presented in an essentially new way it is also of interest to experts in dynamical systems.

Each of the four parts of the book can be the base of a course roughly at the second year graduate level. They are accessible to students having taken standard US first year courses in analysis, geometry and topology. In fact, the background material beyond multivariable calculus and linear algebra and ordinary differential equations is covered in appendices. This allows to use certain parts of the book, especially parts 1 and 3, as the basis for more elementary courses starting from advanced undergraduate (junior or senior) level. Many courses dedicated to more specialized topics can be tailored from this book, such as variational methods in classical mechanics, hyperbolic dynamical systems, twist maps and applications, introduction to ergodic theory and smooth ergodic theory, the mathematical theory of entropy.

In the US any university with a graduate program as well as good undergraduate institutions would be able to thus use the book. In continental Europe the book is appropriate for courses to students at any level above undergraduate, as well as to undergraduate students specializing in mathematics.

This book has been used for courses at institutions world-wide. It is among the 50 most cited mathematics books, and virtually every 21st-century PhD in dynamical systems has been trained using it.

You can read
more about the book.

The picture shows the authors in Oberwolfach
(1997). Photograph by Krystyna Kuperberg.