`UK Nonlinear News`,
`November 2000`

Contributions by L.A. Bunimovich, S.G. Dani, R.L. Dobrushin, M.V. Jakobson, I.P. Kornfeld, N.B. Maslova, Ya.B. Pesin, Ya. G. Sinai, J. Smillie, Yu.M. Sukhov, A.M. Vershik.

459 Pages ISBN: 3-540-66316-9

Encyclopedia of Mathematical Sciences, Vol 100. Springer (2000)

This book provides and extensive and detailed survey of various areas of Dynamical Systems. It contains contributions by several people all of whom have played a central role in the development of the subject over the last thirty years. As Sinai observes in the preface,

``Each author who took part in the creation of this issue intended, according to the idea of the whole edition, to present his understanding and impressions of the corresponding part of ergodic theory or its applications''

This approach works particularly well here as there is a feeling of how the different areas have deep interconnections but their own distinct set of ideas, problems and techniques. These may for example have a more algebraic or analytical slant, or be more directly influenced by problem in Mathematical Physics, and in this respect attract different kind of people according to individual taste.

The style is generally readable although it is not, and is not
meant to be, a pedagogical text. It is rather, ```according to the
idea of the whole edition`'', an ``Encyclopedia''. It contains precise
definitions and statements of results as well as very large numbers of
bibliographic references. Nevertheless it never feels dry and the
presentation generally succeeds in clarifying the relative significance
of various results and ideas in the development of the subject as a
whole.

I would say that it is an invaluable resource for students and other people who are beginning to do research in a particular field and for those who work in related fields and need a synthetic but complete and precise overview of the definitions and results in one of the areas covered here.

The book is divided into 5 parts:

- General Ergodic Theory of Groups of Measure Preserving Transformations;
- Ergodic Theory of Smooth Dynamical Systems;
- Dynamical Systems on Homogenous Spaces;
- The Dynamics of Billiard Flows on Rational Polygons;
- Dynamical Systems of Statistical Mechanics.

Each part is almost a book in itself, divided into several chapters sometimes written by different contributors.

The first part contains an exposition of the basic principles, problems, and results of Ergodic Theory. It helps to put into perspective and motivate the results of the later chapters by sketching the general point of view which has played an important role in shaping the subject. It also contains a survey of the so-called Entropy Theory which contributed many fundamental notions in the 50's and 60's.

The second part contains three main chapters and is the one closest to my particular interests. The first chapter, by Pesin, is an overview of the abstract theory of Hyperbolic Dynamics which was developed first in the Uniformly Hyperbolic setting by Anosov and Smale in the 60's and in the Non-Uniformly Hyperbolic setting by Pesin himself in the 70's. The whole theory is essentially a broad generalization of the Invariant Manifold Theorem for hyperbolic fixed points, where properties of the differential map are shown to have important consequences for the geometrical properties of the actual system. In particular, suitable (hyperbolicity) assumptions on the differential map, imply a subtle geometrical structure of stable and unstable leaves which can be used to investigate many properties of a system, including ergodicity, existence of invariant measures, decay of correlations etc. For a more pedagogical and detailed exposition of this theory see [2]. Chapters by Bunimovich and Jakobson then survey the application of the uniform and non-uniform theory respectively to specific classes of examples, in the latter case mainly restricted to the one-dimensional setting. For a more detailed exposition of the uniformly hyperbolic theory see [3] and [5]. For a comprehensive survey of one-dimensional dynamics see [4] and for more recent developments see [1], and especially in the higher dimensional context see [6].

The third part by Dani is a tight and comprehensive survey of results on a specific class of dynamical systems which arise naturally from translations and affine automorphisms of quotient spaces of Lie groups. There have been great developments in the last ten to fifteen years and these are very well presented here. Links to the area of Diphantine approximations and geometry are also discussed.

The fourth part is a chapter by Smillie on a very special class of
Billiard flows. Here the billiard tables are assumed to be polygons
and this gives rise to a variety of special properties of the flow
which simplify the picture on the one hand but also lead to
connections with other objects such as a class of flows on moduli
spaces which ```are of great dynamical interest`''.

The fifth and final part is concerned with mathematics
motivated by the fact that to understand certain kinds of systems
```consisting of a large number of particles`''
such as the behaviour of
gas or fluids, one is particularly interested in ```results in which
all estimates are uniform with respect to the number of degrees of
freedom`''. Thus ```the fundamental mathematical approach to this
problem is the explicit consideration of infinitely-dimensional
dynamical systems arising as the limit, as N tends to infinity, of the
system of equations of motions of N particles`''. This part contains a
general chapter on Gibbs measures and Interacting Particle Systems,
and a more focussed chapter on the basic existence and uniqueness
theorems for the Boltzmann equations.

On the whole I think that this is an essential text for any library. It would make an excellent addition to a private collection as well I can almost guarantee that if you had it on your shelves you would be using it often as a reference for the exact statement of such and such a theorem or the exact definition of such a notion. If the particular area is quite familiar it also provides a very nice account of the historical and logical development of the area.

**References**:

- V. Baladi. Decay of correlations in dynamical systems. World Scientific (1999)
- Pollicott. Lectures on Ergodic Theory and Pesin Theory on compact manifolds. LMS lecture notes. Vol 180 (1993).
- de Melo, Palis. Geometric Theory of Dynamical Systems. Springer-Verlag (1982).
- de Melo, van Strien. One dimensional dynamics. Springer-Verlag (1993).
- C. Robinson, Dynamical Systems. CRC Press.
- Viana. Stochastic properties of deterministic systems.
See preliminary version at
`http://www.impa.br/~viana`

A listing of books reviewed in `UK Nonlinear News`
is available.

`UK Nonlinear News` thanks
Springer-Verlag for
providing a review copy of this book.

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Page Created: 31st October 2000.

Last Updated: 31st October 2000.

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