Preface
This book is a revised and considerably expanded version of our book Lya-
punov Exponents and Smooth Ergodic Theory [7]. When the latter was
published, it became the only source of a systematic introduction to the
core of smooth ergodic theory. It included the general theory of Lyapunov
exponents and its applications to the stability theory of differential equa-
tions, nonuniform hyperbolicity theory, stable manifold theory (with em-
phasis on absolute continuity of invariant foliations), and the ergodic theory
of dynamical systems with nonzero Lyapunov exponents, including geodesic
flows. In the absence of other textbooks on the subject it was also used as
a source or as supportive material for special topics courses on nonuniform
hyperbolicity.
In 2007 we published the book Nonuniform Hyperbolicity: Dynamics
of Systems with Nonzero Lyapunov Exponents [9], which contained an up-
to-date exposition of smooth ergodic theory and was meant as a primary
reference source in the field. However, despite an impressive amount of
literature in the field, there has been until now no textbook containing a
comprehensive introduction to the theory.
The present book is intended to cover this gap. It is aimed at gradu-
ate students specializing in dynamical systems and ergodic theory as well as
anyone who wishes to acquire a working knowledge of smooth ergodic theory
and to learn how to use its tools. While maintaining the essentials of most
of the material in [7], we made the book more student-oriented by carefully
selecting the topics, reorganizing the material, and substantially expanding
the proofs of the core results. We also included a detailed description of es-
sentially all known examples of conservative systems with nonzero Lyapunov
exponents and throughout the book we added many exercises.
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