This book deals with the general theory of flows on three-dimensional compact manifolds. The authors give the following definition for an attractor – a set of points whose forward trajectories remain inside a bounded region of space forever and such that all nearby trajectories converge to it. The motivation for this book is the recent establishment by Warwick Tucker of the existence of the Lorenz attractor rigorously with a computer-assisted proof. It is interesting that the Lorenz attractor for three dimensional systems is both chaotic and robust.
More precisely, the authors first present an overview of the results of uniformly hyperbolic dynamical systems and some standard generic properties of flows in the \( C^{1} \) topology. The singular horseshoe of Labarca-Pacifico and the geometric Lorenz attractor of Afraimovich-Bykov-Shil’nikov and Guckenheimer-Williams are described in detail. Then they prove that a robustly transitive vector field \( X \) on a three-dimensional manifold is globally hyperbolic. In Chapter 5 robustly transitive sets with singularities are characterized as partially hyperbolic attractors.
The authors introduce the following definition of a singular-hyperbolic set – it is a compact partially hyperbolic invariant subset with a volume hyperbolic invariant set. Sufficient conditions for a singular hyperbolic attractor to be robust are derived. A physical measure with non-zero Lyapunov exponents and positive entropy is constructed by the authors for singular hyperbolic attractors. In this sense a physical measure with non-zero Lyapunov exponents can be considered as another aspect of chaotic behavior. It is shown that singular hyperbolic attractors either have a zero volume or the flow is globally hyperbolic by means of the assumptions of singular hyperbolicity. The Omega-limit set for \(C^1\) generic flows is described and then the global dynamical dichotomy is proved. Finally, the author present many results on the dynamics of flows on three-dimensional manifolds.
As a conclusion, the book under consideration is interesting and presents many results on the dynamical behavior of three dimensional flows with a lot of graphical illustrations which make it more readable.