Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolic
This short volume packs a lot of material into its six chapters. The book is a graduate text and is written in a traditional theorem-proof style, which can be dry--but certainly is concise and precise. The organizing principle is hyperbolicity, and the dynamical systems considered are diffeomorphisms on a metric space. Before attacking this text, a reader will need a knowledge of topology at the level of Munkres’ text and some familiarity with Riemannian geometry. For example, aquaintance with concepts such as Tychonoff’s theorem on the product topology, Finsler structure and Grassmann spaces are helpful. The introductory chapter defines important concepts like omega-limit sets, chain recurrent sets, minimality, transitivity, and topological conjugacy. Moreover, fundamental results such as the Conley decomposition are proved, and this all takes place in some twenty pages! I think that this would be hard going for a student new to the field.
On the other hand, compete proofs of many important results, such as the Hartman-Grobman theorem on topological conjugacy in the neighborhood of a hyperbolic fixed point, the stable manifold theorem, and the structural stability of Anosov toral automorphisms, are given. There are extensive discussions of ideas such as Smale’s horseshoe, Axiom A systems, shadowing, structural stability, Omega-stability, and Smale’s decomposition theorem. A nice feature of the text is that there is a strong parallel drawn between theorems for hyperbolic fixed points and more general hyperbolic sets: once the former are understood, then the proofs of the latter follow quickly. The proofs are carefully constructed and contain all the needed details. The final chapter introduces the concept of “quasi-hyperbolicity”: a compact invariant set for which all nonzero orbits of the tangent map are unbounded in at least one direction of time.
This book has a limited goal, and thus some important ideas do not find a place. For example, while “chaos” is defined in Chapter one (Devaney’s notion of transitivity and dense periodic points which implies—as is proven in the book—sensitive dependence), but the discussion is very brief. Smale’s horseshoe construction is explained and the topological conjugacy to the two-shift is shown, but the Smale-Birkhoff theorem on dynamics near a transverse homoclinic point is only stated. Similarly, Palis’ Lambda-lemma is stated but not proven, and there is no discussion of the breakdown of hyperbolicity, for example, by the Newhouse mechanism. Finally, ideas such as non-uniform hyperbolicity, Lyapunov exponents, SRB measures, Pesin theory, and Oseledec’s multiplicative ergodic theorem are not discussed, nor are the results of Benedicks and Carleson on the Hénon attractor. Nevertheless, the discussion of many concepts for hyperbolic dynamics is complete, proceeding from basic definitions through to the final results.
There are also many carefully thought-out exercises that would make this a good text for an advanced student, and the care with which the results are stated and proven makes this an excellent reference for a practitioner.