This is a graduate text in differentiable dynamical systems. It
focuses on structural stability and hyperbolicity, a topic that is
central to the field. Starting with the basic concepts of dynamical
systems, analyzing the historic systems of the Smale horseshoe, Anosov
toral automorphisms, and the solenoid attractor, the book develops the
hyperbolic theory first for hyperbolic fixed points and then for
general hyperbolic sets. The problems of stable manifolds, structural
stability, and shadowing property are investigated, which lead to a
highlight of the book, the $\Omega$-stability theorem of
Smale.
While the content is rather standard, a key objective of the book
is to present a thorough treatment for some tough material that has
remained an obstacle to teaching and learning the subject matter. The
treatment is straightforward and hence could be particularly suitable
for self-study.
This book introduces the reader to some
basic concepts of hyperbolic theory of dynamical systems with emphasis
on structural stability. It is well written, the proofs are presented
with great attention to details, and every chapter ends with a good
collection of exercises. It is suitable for a semester-long course on
the basics of dynamical systems.
—Yakov Pesin, Penn State University
Lan Wen's book is a thorough introduction to
the “classical” theory of (uniformly) hyperbolic dynamics,
updated in light of progress since Smale's seminal 1967 Bulletin
article. The exposition is aimed at newcomers to the field and is
clearly informed by the author's extensive experience teaching this
material. A thorough discussion of some canonical examples and basic
technical results culminates in the proof of the Omega-stability
theorem and a discussion of structural stability. A fine basic text
for an introductory dynamical systems course at the graduate level.
—Zbigniew Nitecki, Tufts University
Readership
Graduate students and research mathematicians
interested in the hyperbolic theory of dynamical systems.